Classnotes for APM 581 Geometry and Control of Dynamical …kawski/classes/apm581/10fall/... ·...

Classnotes for APM 581

Geometry and Control of Dynamical Systems I

Matthias Kawski

Arizona State University

http:\math.asu.edu~kawski

Fall 2010.

Chapter 1: December 18, 2010

These course-notes are work in progress and build on earlier versions of class-notesin differential geometry and nonlinear control theory. Whereas for either part thereare many suitable references, one goal is for these notes to eventually grow intoa comprehensive set that addresses in a unique way this intersection of classicaldifferential geometry with geometric control theory.Many sections of the 2009 version are very close to earlier versions for differentialgeometry, which originated from a first type-set draft prepared for a course in spring2000 at ASU. These are very close to (but nowhere as accurate) as Spivak’s books– and they were justified only by Spivak being out of print when the 2000 springsemester began. Since then Spivak’s volumes have been republished (now typeset,no longer type-written), and every user of these notes is expected to eventually buythe originals by Spivak – he deserves his royalties!

Much of the organization and notation go back to notes taken in a class taught byAl Lundell at the U. of Colorado in 1983/84, and on two classes taught in 1989 and1991 at ASU. Originally, they may have been quite independent, but upon efforts tomake them more comprehensive, more precise, they again converged ever more toSpivak’s treatment. However, there are various places where the current notes moreclosely follow Sternberg (e.g., notation and terminology involving tensors), Boothby(e.g., Riemannian basics), Marsden, and others. The main practical value of thedifferential geometry notes is that they use the same notation (even if it is just ui

in place of xi) that the instructor has become too accustomed to, and will use inclass... and they selected integrate questions / exercises that match this class.

The current version includes more diagrams to improve the readability. The reviewsof basic topology and differentiability, which really belong into an appendix, areincluded in the order that the class actually covered them. Accompanying explo-rations using computer algebra systems and other computing technology have beenmoved further into the background.

The parts addressing control theory are loosely oriented about the classical text-books by Isidori and Nijmeier / van der Schaft (classical control topics), and recentmonographs by Bloch and Bullo / Lewis (mechanical applications and affine con-nections), and Agrachov / Sachkov (advanced geometric points of view).

http:\math.asu.edu~kawski

Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. December 18, 2010 6

1 Revisiting curves in the plane and 3-space

The main objectives of this first section are: (i) begin to sensitize for, and develop the concept ofgeometric properties, (ii) establish a counterpoint for later to be developed intrinsic properties,(iii) raise the first questions about the nature of first and second derivatives (where do theylive?), and (iv) demonstrate the desired level of precision of language (emphasizing functions).The technical setting should be mostly familiar from elementary multi-variable calculus.

1.1 Definition of a curve

In many settings it may be appropriate to think of a curve as a set of points in the plane orin 3-space. However, in differential geometry and other advanced settings, it is generally moreconvenient to work with a different notion – basically calling what previously was named aparameterization the “curve”.

Definition 1.1 A curve is a continuous function defined on an interval J ⊆ R, taking values ina (topological) space M (in this section M is assumed to be Rn). (The interval in this definitionmay be open or closed, finite, semi-infinite or the entire real line. At this time only assumeenough structure on the space M so that one can talk about continuity.)

The key difference is that with this definition a curve is a function. Consequently it has a richerstructure than just a set of points – a structure that facilitates technical analysis. This definitionalso easily carries over to much more general settings – e.g. think of a vibrating membrane as acurve in an appropriate space M of functions of two variables. What matters is that the spacehas enough structure (at least a topology) so that one may talk about continuity. (Upcomingwork will require additional structures on the space M so that one can differentiate curves.)A curve γ : J 7→ M is called closed if J = [a, b] is a (finite) closed interval and γ(a) = γ(b). Ifthe restriction of a closed curve γ to [a, b) is one-to-one, then γ is called a simple closed curve.

Example 1.1 The circle S1 = {(x, y) ∈ R2 : x2 + y2 = 1} is the image of the simple closedcurve γ : [0, 2π] 7→ R2 defined by γ(t) = (cos t, sin t). (Note that the circle S1 is not a curve!)

Figure 1: Reparameterization of a curve

Definition 1.2 If γ : J 7→ M is a curve defined on a finite interval J and φ : J 7→ I is acontinuous function that maps a finite interval J onto I (mapping endpoints to endpoints), thenthe curve σ = γ ◦ φ : I 7→M is called a reparameterization of γ.

The notion of reparameterization may be extended to infinite intervals provided one suitablymodifies the notion of endpoint to endpoint, e.g. requiring the existence of the limits limt →±∞and that these equal −∞ and ∞. Among one-to-one reparameterizations one distinguishesorientation-preserving and orientation-reversing reparameterizations according to whether themap φ maps the left endpoint of J to the left or right endpoint of I.


1.2 Differentiable curves and arc-length

An intuitive notion of the length of a curve in Rn may be built on successive approximationsby polygonal approximations. More specifically, suppose that γ : [a, b] 7→ Rn. Let ‖ · ‖ denotethe Euclidean norm ‖(x1, . . . xn)‖ =

√x21 + . . . x2n in R

n. Define the length L(γ) of the curve asthe supremum (possibly infinite)

L(γ) = supP

n(P)∑i=0

‖γ(ti+1)− γ(ti)‖ (1)

where P ranges over all partitions P = {ti : 0 ≤ i ≤ n(P)} such that a = t0 < t1 < . . . tn(P) = b.It is important to note that this definition of length cannot directly generalize to spaces forwhich one does not have an a-priori notion of distance – i.e. where ‖ · ‖ has no meaning (yet).A key observation is that for continuously differentiable curves there is a natural alternative:the length is the integral of the speed. This notion generalizes, even give rise to the concept ofRiemannian manifolds. Loosely speaking, the main idea is to rewrite

n∑j=0

‖γ(ti+1)− γ(ti)‖ =n∑j=0

‖γ(ti+1)− γ(ti)‖(ti+1 − ti)

· (ti+1 − ti) −→∫ ba‖ γ′(t) ‖ dt =: L(γ) (2)

By fairly straightforward (advanced) calculus arguments one may make this idea rigorous, i.e.show that for any continuously differentiable curve defined on a finite closed interval there existsa unique limit which defines the length of the curve. Recall the definition of the derivative forcurves that take values in Rn:

Definition 1.3 A curve γ : (a, b) 7→ Rn is called differentiable if for every t ∈ (a, b) the limitlimh→0 1h (γ(t+ h)− γ(t)) exists. If the limit exists, it is denoted γ

′(t) and called the velocity atγ(t) (or at t).

This notion naturally generalizes to infinite dimensional normed linear spaces. But the subse-quent work goes in a different direction: The initial main focus is to develop a natural notionof a tangent vector to a curve (taking values in an abstract manifold) that does not require anyprior notion of a difference γ(ti+1) − γ(ti) of two points in that space. Once one has such ageneralized notion of a tangent vector, much of the following fundamental notions, structures,calculations and arguments will carry over to the general case of abstract manifolds.

Aside, on a more advanced level, a key idea is to make sense of such differencesby interpreting objects such as points as linear functionals on the space of smoothfunctions on the manifold – material for the next class! As a first reference, in thesetting of control, consult A. Agrachëv and Yu. Sachkov, “Control Theory from aGeometric Viewpoint”, (2004), Springer.

The second derivative γ′′(t) is defined analogously, and is called the acceleration at γ(t) or at t.The magnitude ‖γ′(t)‖ of the velocity is called the speed. In the case of plane and space curves oneroutinely identifies the point γ(t) ∈ Rn with the arrow (vector) from the origin to this point. Onthe other hand the velocity and acceleration are commonly visualized as arrows (vectors) rootedat the point γ(t), or even at γ(t) + γ′(t). There appears to be a certain arbitrariness about thisrepresentation – but it seems to make sense after a little thought. The upcoming constructionof the tangent bundle will illuminate the situation and provide clarifying distinctions. A helpful


preparation at this time is to think about possible alternative representations, and to find goodarguments why the usual placements of the arrows are a good choice without any compellingalternative. Also think of how these arrows are affected by changes of units, e.g. going frominches to centimeters, or from minutes to seconds.

Exercise 1.1 Show that if the velocity γ′ is constant then the (image of the) curve γ is a straightline, but the converse is not true.

Exercise 1.2 Verify that the curve γ : R 7→ R2 defined by γ(t) = (cos(t), sin(t)) has constantspeed, but nonzero acceleration.

Exercise 1.3 Prove that the acceleration γ′′ is orthogonal to the velocity γ′ (for all t), i.e.〈γ′(t) , γ′′(t)〉 ≡ 0 if and only if the speed is constant.(Hint: Differentiate c ≡ ‖γ′(t)‖2 = 〈γ′(t) , γ′(t)〉. Read the identities both directions.)

Exercise 1.4 Verify that the curve γ : R 7→ R2 defined by γ(t) = (t2, 0) if t ≥ 0, and γ(t) =(0, t2) if t < 0 is continuously differentiable, yet its image in the plane has a corner.

Blanket assumption: For most of the following assume that all curves underconsideration are twice continuously differentiable and that ‖γ′(t)‖ 6= 0 for all t.

In many cases one may conveniently assume that the curve is smooth, i.e. that it has continuousderivatives of all orders. Note that the assumption that the speed is never zero eliminatessuch nuisances as the corners exhibited by the continuously differentiable curve of exercise 1.4.Moreover, it also prohibits such nuisances as exhibited by the curve γ(t) = (cos(t−t3), sin(t−t3))which “go back and fourth” along the image of the curve. If there is a need to allow for suchbehaviors it is usually easy to either consider the curve piecewise, or relax the requirements inspecific cases and then adapt the desired theorems as needed.

Obviously, different curves may have the same image. This corresponds to an arbitrarinessof choosing a specific parameterization by which to represent the (image of the) curve. Fortheoretical purposes one generally prefers to work with a canonical reparameterization of adifferentiable curve γ. A natural such choice is so that the parameter t (often considered astime) agrees with the distance traveled along the curve.

Definition 1.4 The arc-length of a continuously differentiable curve γ : [a, b] 7→ Rn is definedas the function ψ : [a, b] 7→ R (commonly one writes s = ψ(t)),

ψ(t) =∫ ta〈γ′(τ) , γ′(τ)〉1/2 dτ (3)

The curve γ is called parameterized by arc-length if ψ(t) ≡ t, or, equivalently, 〈γ′ , γ′〉 ≡ 1.

This definition relies on the standard inner product 〈· , ·〉 which in Rn is almost synonymouswith the notion of (Euclidean) distance. The key idea underlying Riemannian geometry (com-pare chapter 4) is that once one has a suitable generalization of this inner product, then mostof the notions and properties naturally carry over to abstract settings.To explicitly reparameterize a given curve γ : [a, b] 7→ Rn by arc-length generally requiresone not only to evaluate the integral in equation (3) in closed form, but in addition, to solvethe equation s = ψ(t) for t in terms of s – generally a hopeless task if looking for closed


form expressions in terms of the traditional elementary functions. Thus one usually has tochoose between either formal expressions (for theoretical purposes), and numerical techniques(in practical calculations).Provided one has a notion of length in the space, the parameterization by arc-length is natural inthe sense that it avoids arbitrary choices. But the cost of complicated or intractable expressionsin most practical examples is more than compensated for by the elegant descriptions of thegeometry of the curve when utilizing parameterizations by arc-length.

Exercise 1.5 Reparameterize the curve γ : [0, 1) 7→ R2 defined by γ(t) = (√

1− t2, t) by arc-length by explicitly integrating equation (3) and solving for t in terms of s.

Exercise 1.6 Calculate and sketch the graphs of the arc-lengths of the curves γ1(t) = (t, t),γ2(t) = (cos 2πt, sin 2πt), γ3(t) = (t, t2), and γ4(t) = (cos 2πt, sin 2πt, c t), all defined for t ≥ 0.If feasible, reparameterize each curve by arc-length.

Exercise 1.7 Verify by direct calculation that arc-length of plane curves is invariant underorthogonal linear transformations: More specifically, let γ : [a, b] 7→ R2 and σ : [a, b] 7→ R2 betwo differentiable curves that are related by σ = A · γ where A satisfies AAT = I. Explain ingeometric terms – e.g. using the earlier definition in terms of polygonal approximations – whythis is expected.


1.3 Curvature of plane and space curves

In a very general sense, differentiability makes precise the intuitive idea of being approximableby a linear object, think of tangent lines and planes, or more generally by linear functions andmaps. Curvature may loosely be thought of as a quantification how far the object is from locallybeing linear. Curvature may well be considered as the central concept in differential geometry.In the case of graphs of functions y = f(x) all calculus students learn that the second derivativeis somehow related to how much the graph curves, by, in some sense, measuring how far the firstderivative is from being constant. But any quick look at the graphs of polynomial functions ofdegree at least two illustrates that the second derivative does not well correspond to the intuitivesense of eventual straightness of their graphs.

Exercise 1.8 Sketch the graphs of the polynomial functions p : x 7→ x2 and q : x 7→ x3, oftheir second derivatives, and of the curvature of their graphs (as functions of x, see below for astandard formula). Discuss the respective limits as ‖x‖ → ∞ in view of the shape of the graphs.

This first shortcoming of using the second derivative for quantifying curvature is confounded bya lack of being geometric, in the sense of not being invariant under basic transformations,. asillustrated in the next exercise.

Exercise 1.9 Consider the graph of f0(x) = x2 for |x| ≤ a. Verify that for small values ofa > 0 and for small angles θ ≤ θmax(a) the image of the graph under a rotation by an angle θabout the origin is again the graph of a function fθ(x). Find an explicit formula for fθ and showthat its second derivative f ′′θ is not constant equal to f

′′0 ≡ 2.

Suggestion: Use (x, y) to denote points on the original curve y = x2 and let (ξ, η) denote pointson the rotated curve. Express x and y in terms of ξ and η (compare exercise 1.7), substituteinto y = x2 and solve for η in terms of ξ. Finally calculate d

2ηdξ2

.

Two culprits come to mind: First, the derivatives in the two preceding exercise are taken withrespect to the first coordinate x, as opposed to the intrinsic arc-length parameter. Secondly,curvature should measure rate of change of direction, which is not the same as rate of changeof slope, the distortion being due to the tan y′ = tanα.

In the following T,N, σ, κ, . . . are the correct function names. However, for emphasis,the discussion here will often write T (s), N(s), κ(s), . . . etc. This will help distinguishthese from the compositions T ◦ ψ, κ ◦ ψ, . . . which with common abuse of notation, oftenare written as T (t), κ(t), . . .. The reader is encouraged to explore even working with thefunctions (in precise notation) T ◦ σ−1, κ ◦ σ−1, . . . whose common domain is the image ofthe curve. It will become clear that is is usually more convenient to consider T,N, κ, . . .as functions defined on the parameter interval I.

Consider smooth curves σ : I 7→ R2 and σ : I 7→ R3 that are parameterized by arc-length, i.e.,‖σ′‖ ≡ 1. At any s ∈ I, the velocity is a unit tangent vector to the curve, suggesting the notationT (s) for σ′(s). In the case of plane curves it is convenient to immediately complete, for everys ∈ I, {T (s)} to a positively oriented orthonormal basis (frame) {T (s), N(s)}.In general, to describe the rate of change of this direction T = σ′ differentiate again. Recallfrom exercise 1.3 that 〈σ′ , σ′〉 ≡ 1 implies that σ′′ ⊥ σ′. Define the curvature κ and (principal)normal N as the magnitude and direction of σ′′, i.e. use as σ′′ = κN and 〈N , N〉 = 1 (together


with κ ≥ 0) as the defining equations for both κ and N . Note that N is only defined for thoses ∈ I such that κ(s) 6= 0. Caveat: In the case of plane curves it is often convenient to allow thecurvature to be the signed magnitude, and instead demand that {T,N} has positive orientation.Similarly, in 3-space, it is convenient, before any further differentiating, to complete the pair{T,N} to a positively oriented orthonormal three-frame {T,N,B} by defining for each s ∈ Iwhere κ(s) 6= 0 a third unit vector B(s) = T (s) × N(s) (using the standard cross-product in3-space). Note the analogy to selecting N in the planar case as the last vector to completean orthonormal frame – and allowing the coefficient to have both positive and negative values.This also suggests an obvious generalization to higher dimensions n > 3. The triple (T,N,B)is called the Frenet frame along the curve σ. It is a function defined at all points s ∈ I whereκ(s) 6= 0), with values conveniently interpreted as 3× 3 matrices.To continue the investigations differentiate N By an argument analogous to the one employedearlier, N ′ = ddsN is orthogonal to N , and hence may be written as a linear combinationddsN(s) = a21(s)T (s) + τ(s)B(s). The function τ is called the torsion of the curve, whichintuitively, quantifies the rate at which the curve twists out of a plane – compare the exercise1.14. To identify the parameter a21, differentiate the identity 0 ≡ 〈T , N〉 and find that

0 ≡ κ〈N , N〉+ a21〈T , T 〉+ τ〈T , B〉.Since the third term vanishes, it is clear that a21 = −κ. To complete the analysis differentiatethe identities 0 ≡ 〈T , B〉, 0 ≡ 〈N , B〉, and 1 ≡ 〈B , B〉 to obtain B′ = −τN . Taken together,these differential equations form the Frenet-Serret formulas

T ′ = κNN ′ = −κT τBB′ = −τN.

(4)

To emphasize the characteristic structure of this set of equations, rewrite it by forming a matricesΩ = (T |N |B ) and Ω′ = (T ′ |N ′ |B′ ) whose columns are the representations of these vectorswith respect to the standard coordinates in R3. Note that Ω is an orthogonal matrix, i.e.ΩTΩ = ΩΩT = I3×3. With this notation, the Frenet-Serret formulas become

Ω′ = ΩA where A =

0 −κ 0κ 0 −τ0 τ 0

(5)Preview: Later, Ω will be considered as a point (!) in the three dimensional manifold (actuallya Lie group) SO(3) of 3 × 3 orthogonal matrices. The product ΩA, which again is a 3 × 3orthogonal matrix (though generally not orthogonal) will be considered a tangent vector at Ω,and it will be clear from general considerations (in some sense generalizing the arguments madeabove about the derivatives of the inner products between the columns of the matrix Ω) why Ahas to be skew symmetric, i.e. AT = −A .

When given an initial reference frame Ω(0) consisting of three orthogonal unit vectors T (0), N(0)and B(0) together with two sufficiently regular functions κ(s) and τ(s) this system of differentialequations uniquely determines Ω(s) for all times s, considering Ω as a curve in SO(3).Continuing further, if in addition an initial point σ(0) ∈ R3 has been specified, then the FrenetSerret formulas together with the differential equation σ′ = T (s) uniquely determine a curvein R3. It is straightforward to verify that the curvature and torsion of this curve agree withthe data provided to the differential equation. A corollary of this study is that the curvature(if everywhere nonzero) and torsion completely determine a smooth curve up to translation, asdetermined by σ(0), and rotation, as determined by Ω(0).


Exercise 1.10 Explain how this gives as a corollary that curvature and torsion are invariantunder translation and rotation (compare exercise 1.7).

Exercise 1.11 Explore how complicated a brute force linear algebra calculation is (similar toexercise 1.7) that directly shows that curvature and torsion are invariant under rotations andtranslation. (It may be appropriate to use a computer algebra system for part of this work.)

Exercise 1.12 Show that if the curvature κ ≡ 0 of a plane curve vanishes identically, then thecurve is a straight line. Is the same true for a curve in 3-space? Explain!

Exercise 1.13 Show that if the curvature κ ≡ c 6= 0 of a plane curve is constant, then the curveis a circle with radius 1/c. Is the same true for a curve in 3-space? Explain!

Exercise 1.14 Show that if the torsion τ ≡ 0 of a space curve vanishes identically, then thecurve lies in a plane.

1.4 A few remarks of curvature of curves as control (in progress)

A different way of looking at the differential equation T ′ = κ is to consider κ as a control thatshall be chosen so that the curve has desired properties and which possibly is optimal in asuitable sense. In a simple example, the curve represents a path along which a robot, a boat,insect, or an aircraft that is approaching an airport, moves. Commonly, the speed of the boatmight be fixed (normalized to equal one), and the control corresponds to steering the boat. Inmany applications it is natural to impose a constraint on the size of this control, correspondingto a bound on the curvature. The first problem is to find possible controls, that is curvaturefunctions, such that the corresponding curve connects a given initial to a desired terminal point(and orientation) Additional objectives may be to do this in minimum time, which is the same asminimal length of the curve, or minimizing the total curvature

∫ L0 κ ds or minimizing

∫ L0 κ

2 ds.In the case of the landing approach of an aircraft, a generalization combines this with a dynamicinterpolation problem that demands that the plane passes through a sequence of predeterminedcontrol points. This course plans to revisit some of these problems later in the setting of geodesicsand optimal control. This is just an opportune point to not only consider the curvature of agiven curve, but instead begin reflecting on the meaning and usefulness of the inverse questions.Some names associated with these problems are the Dubins problem and the Reeds Shepp car.Adding a drift term, think wind or current, one has Zermelo’s navigation problem. Some typicalresults prove that the optimal solutions exist and that they are concatenations of straight linesand circles, or helices. Recent work revisits these problems for paths that are constrained to lieon curved surfaces. A few selected references to get started are• L. Dubins, On curves of minimal length with a constraint on average curvature and with

prescribed initial and terminal positions and tangents, Amer. J. Math. 79 (1957) 497-516.

• M. Sigalotti, and Y. Chitour, Dubins’ problem on surfaces. II. Nonpositive curvature,SIAM J. Control Optim. 45 (2006) 457–482.

• H. Sussmann The Markov-Dubins problem with angular acceleration control, Proc. 36thIEEE Conf. Decision and Control (1997) 2639-2643.

• U. Serres On the curvature of two-dimensional optimal control systems and Zermelo’snavigation problem J. Mathematical Sciences 135 (2006), 3224–3243.


1.5 Practical calculations for explicit examples

The Frenet-Serret formulas provide a beautiful and comprehensive geometric description of thecurves in 3-space. They appear to intrinsically rely on parameterizations by arc-length, yet formost curves explicit closed-form formulas for parameterizations by arc-length are beyond reach.However, note that all these formulas only involve derivatives of the curve σ. Consequently,there is no need to ever explicitly calculate the arc-length. All that is needed is the integrand ofthe formula (3) – the chain-rule does the rest.Consider a smooth curve γ : J = [a, b] 7→ Rn. Define ψ(t) =

∫ ta

√〈γ′(τ) , γ′(τ)〉 dτ . The usual

assumption ‖γ′(t)‖ > 0 for all t ∈ J guarantees that ψ has an inverse, which is again smooth.The curve σ : I = [0, L(γ)] 7→ Rn defined by σ = γ ◦ ψ−1 is the reparameterization of γ byarc-length.Differentiating γ = σ◦ψ, in traditional notation ddt(γ(t)) =

ddt(σ(s(t))), the chain rule relates the

velocities γ′ = ψ′ ·(σ′ ◦ψ) = ‖γ′‖ (T ◦ψ) In words, for each t ∈ J , the velocity vector v(t) = γ′(t)points in the same direction as T (ψ(t)) but it generally has non-unit magnitude (or “speed”)‖γ′(t)‖. In practical calculations one typically first obtains γ′, then ‖γ′‖ and T . The key toavoiding excessively unpleasant calculations is to never differentiate normalized expressions suchas T or N , but rather first take suitable cross- and dot-product of derivatives of γ. The nextstep is to note that γ′′ = (σ′′ ◦ ψ) (ψ′)2 + (σ′ ◦ ψ)ψ′′ implies that for every t ∈ J the vectorγ′′(t) lies in the plane spanned by T (ψ(t)) and N(ψ(t)), called the osculating plane. In practicalcalculations in 3-space one calculates γ′′, then calculates B by normalizing the cross-productγ′ × γ′′. Only afterwards(!) one calculates N = B × T . Returning to the acceleration γ′′, themagnitudes of its tangential and normal components are easily calculated as

a‖ = (T ◦ ψ) · γ′′ =γ′ · γ′′

‖γ′‖and a⊥ = (N ◦ ψ) · γ′′ = ±

√‖γ′′‖2 − a2‖ (6)

The curvature κ (and radius of curvature % = 1κ) and the torsion may be obtained in variousways. Typical formulas suitable for practical calculations (for space curves) are

κ ◦ ψ = | a⊥|‖γ′‖2

=‖γ′ × γ′′‖‖γ′‖3

and τ ◦ ψ = (γ′ × γ′′) · γ′′′

‖γ′ × γ′′‖2(7)

Exercise 1.15 Derive the formulas presented above for κ ◦ψ and for τ ◦ψ from the definitionsof κ and τ in terms of the Frenet formulas.

Exercise 1.16 Consider a smooth planar curve γ : J 7→ R2, not necessarily parameterized byarclength. Devise a practical strategy to calculate T,N, κ with minimal effort. Note, that fromT one easily obtains N by interchanging the components and changing the sign of one of thecomponents. Which one? Why?

Exercise 1.17 For curves in the plane given as graphs of functions, i.e. γ(t) = (t, f(t)) orcasually written y = f(x), derive the usual formula κ = y′′/(1+(y′)2)3/2 for the curvature. Notethat technically the above stands for κ ◦ γ1 ◦ ψ = γ′′2/(1 + (γ′2)2)3/2.

Exercise 1.18 Verify that the curve γ : R 7→ R2 defined by γ(t) = (exp(−1/t2), 0) if t > 0,γ(0) = (0, 0) and γ(t) = (0, exp(−1/t2), 0) if t < 0 is infinitely many times continuously dif-ferentiable on the whole real line. Describe the {T (s), N(s)}-frame for this curve (this is veryeasy), with special attention to what happens when t = 0.


Project 1.19 Write a MAPLE procedure that takes as input a space curve, i.e. algebraic ex-pressions for x(t), y(t) and z(t) (and possibly other parameters such as the domain J), and whichgives as output an animation of the Frenet frame along the curve. As test curves consider a helixγ(t) = (cos t, sin t, t) and the (2,3)-torus-knot γ = u−1 ◦ ` where `(t) = (2t, 3t) for t ∈ [0, 2π] andu−1(θ, φ) = ((R+ r sinφ) cos θ, (R+ r sinφ) sin θ, r cosφ), e.g. for R = 5 and r = 2.

Project 1.20 Consider a family of closed curves γ(s, t) that are parameterized by arc-length sand that evolve with time t according to e.g. the heat-equation ∂

2

∂s2κ(s, t) − ∂∂tκ(s, t) = 0. In-

tuitively, this is the easiest model that describes how loops may try to straighten out under theinfluence of tension.For simplicity, start with initial curvature functions κ(s, 0) that are expressed as (finite) Fourierpolynomials κ(s, 0) = a0 +

∑Nj=1(aj cos jt+ bj sin jt). This allows one to explicitly write out the

solutions κ(s, t) = a0 +∑N

j=1 e−j2t(aj cos jt+ bj sin jt) of the heat equation.

First find conditions on the Fourier coefficients that assure that the associated curve is closed.Then integrate the two-dimensional analogue of the Frenet-Serret formulas to obtain the associ-ated curve σ(t, s). Animate the images of the curves.An exploratory worksheet that addresses this project is available from the WWW-sitehttp://math.asu.edu/̃kawski/MAPLE/MAPLE.html. However, it has at least a cosmetic flaw asit arbitrarily fixes σ(0, t) and ∂∂sσ(0, t) – a nicer solution would include a more physical solution,e.g. fixing the center of mass of the curve by translating the curve as needed. Even better, itwould be nice to add, if necessary, a rotation so that the total angular momentum as determinedfrom ∂∂tσ(t, 0) is constant equal to zero. Any improvements of this worksheet are highly welcome.They likely will lead to further, even more interesting applications – starting with an analogousexploration of loops in 3-space!

1.6 A brief pointer to classic associated curves

A huge amount of work has been done to study various curves associated to a given curve. Mostintimately related to the above review are evolutes and involutes, which are the curve traced bythe centers of curvature σ + 1κN and the original curve, respectively.Closely related are curves traced by a fixed point on one curve as it rolls without slipping arounda fixed curve. Typical examples are cycloids (circle rolls on a line) and epicycloids (a circle rollson a circle). Such associated curves are commonly collectively called pedal curves. This subjectis ideally suited for experimentation and interactive visualization and there are numerous freeJAVA applets available. A classic one is the Famous curves applets from St. Andrews University,available on-line at http://www-history.mcs.st-andrews.ac.uk/Java/.While this course goes into a different direction, it still recommends that the reader familiarizesher/himself with this rich body of classic studies. Many of these curves have modern applications,and they appear at quite surprising places even in very abstract studies.

Exercise 1.21 Explored the free applets available and familiarize yourself with the names, def-initions and concepts behind them, and typical images. Enjoy the beauty.

http://www-history.mcs.st-andrews.ac.uk/Java/


2 Manifolds

2.1 Introduction

Manifolds may be thought of as abstractions and generalizations of the intuitive notions of curvesand surfaces. This subsection discusses key ideas, purposes and examples. The next subsectionsreview fundamental topological notions to prepare for a precise definition of manifolds, first inthe topological category, and then in the differential category.The upcoming definition characterizes a manifold as a space in which every point has a neigh-borhood that is homeomorphic to an open subset of a Euclidean space Rn. In particular, thisdefinition does not allow for edges and boundaries thereby avoiding further technical complica-tions. The next step equips manifolds with differentiable structures that allow for notions suchas continuous-time dynamical systems evolving on the manifolds, and for generalized notions ofcurvature. Typical objectives are to analyze the effects of curvature on the global topologicalstructure or on the behavior of dynamical systems. A need to integrate over (subsets of) man-ifolds arises naturally. A major role of local coordinate charts is to transfer these differential(and integral) concepts back into familiar Euclidean space where standard techniques may beemployed for calculations.

Before proceeding to technical descriptions first take a brief look at some typical examples thatshould be included in any notion of manifold. Curves and surfaces, especially the Euclideanspaces Rn, and (open) subsets of Euclidean spaces should be manifolds. Curves and surfacesmay arise as images (or graphs) of functions defined on open subsets of Euclidean spaces (and,more generally, defined on manifolds). The illustration shows the surface defined by % = sin 2φ ·cos 3θ in spherical coordinates, i.e the image of the rectangle [0, 2π]× [0, π] which edges suitablyidentified under the map F : (θ, φ) 7→ sin 2φ cos 3θ · [cos θ sinφ, sin θ sinφ, cosφ].The characterization of the two dimensional sphere S2 ⊆ R3 as the set of all (x, y, z) ∈ R3 thatsatisfy x2 + y2 + z2 = 1, invites a natural generalization to higher dimensional analogues ofsurfaces as subsets of R that may be characterized by (sets of) equations Fk(x1, x2, . . . xn) = 0(k = 1, . . . p). In other words the manifold is given as a preimage F−1({0}). More generally, fora function a function F : M 7→ N between manifolds and a submanifold P ⊆ N , the preimageF−1({N}) ⊆M may be a submanifold of M .Generally one imposes technical conditions that guarantee that there are no self-intersections,edges or boundaries, or, in the category of differentiable manifolds, cusps or corners. A typical


such condition is that the gradient (or a higher dimensional analogue) of some function does notvanish.As a special case, this description immediately opens the door to objects such as the group ofspecial orthogonal matrices SO(3) in three or SO(n) in n dimensions. The defining equationA>A = In×n is of the same form as the equation of the sphere given above. What makes thesematrix manifolds particularly interesting is their natural group structure – there is a naturalnotion of multiplying points on the generalized surface which is compatible with the topologicalstructure. This is the starting point for Lie groups.A different way that many manifolds of interest arise is as quotients. In the most simple casethe circle S1 arises as a quotient of R by Z. Intuitively, for any periodic function f withperiod p > 0, i.e. f(x + p) = f(x) for all x ∈ R, one may consider as its natural domain anyinterval [a, a + p] with endpoints identified. More abstractly, consider the equivalence relation∼ defined on R by x ∼ y ⇐⇒ (x − y)/p ∈ Z. Then each point on the circle Θ represents anequivalence class [Θ] = {Θ + k p : k ∈ Z}. In an analogous way, the torus arises naturally (e.g.very commonly in dynamical systems) as the quotient of the plane R2 by Z2. One commonlyvisualizes the torus as the unit square [0, 1]× [0, 1] with opposing edges identified.

Picture: squares with identified edges, cylinder, torus, Klein bottle

If one starts with the same square, but identifies one (or two) sets of opposing edges withorientation reversed one arrives at the Klein bottle and at the projective plane. Neither one ofthese can be visualized in the usual way as a surface in R3, but apparently each shares manyproperties with the torus due to their analogous construction. Similar construction quickly getvery challenging: Gluing together different faces of a cube in R3 is still simple, but what aboutidentifying suitable pairs of faces of a dodecahedron? A good start for further reading withcompelling speculations about astrophysics is the article “The Poincaré Dodecahedral Space andthe Mystery of the Missing Fluctuations” by J. Weeks, AMS Notices 51 no. 6 (2004) pp. 610 -619, www.ams.org/notices/200406/fea-weeks.pdf.More abstractly, projective spaces arise as spaces of all lines in Rn that pass through the origin.Before looking at this more closely, recall the simple case of the space of all (semi-infinite, open)rays emanating from the origin. Each of these rays may be naturally identified with the point onthe unit sphere (unit circle) through which it passes. Thus one may think of the spheres Sn−1 asarising from Rn\{0} as quotients under the equivalence relation x ∼ y ⇐⇒ if there exists λ ∈ R,λ > 0 such that x = λy. In a practical sense this is closely related to considering only the angle(s)θ (or (θ, φ)) when working with polar (or spherical coordinates). This is a very simple case of agroup (here (R+, ·) acting on a manifold (here Rn \{0}), the equivalence classes being the orbitsof the group action. For general conditions when a generalization of such quotients (motivatedby realizations of control systems) again yields a manifold see e.g. “A generalization of the closed

www.ams.org/notices/200406/fea-weeks.pdf


subgroup theorem to quotients of arbitrary manifolds” by H. Sussmann, J. Differential Geometry10 (1975), pp. 151–166.In analogy, if one discards the requirement λ > 0 in the preceding definition of equivalence, thenthe equivalence classes are the lines through the origin. (More precisely, since the constructionstarted with Rn \ {0}, the origin is removed from each line, and each equivalence class reallyconsists of two rays.) One may visualize the resulting quotient space as the space of pairs ofopposite points on the sphere Sn−1, or as a semi-sphere with two halves of the equator (which isa sphere Sn−2 by itself) identified, or glued together with careful attention to the orientation ofeach piece.From these projective spaces it is only a small step to Grassmannian manifolds which maybe constructed as spaces of m-dimensional (hyper-)planes through the origin in n-dimensionalEuclidean space. A typical application where these appear naturally is in the classification oflinear control systems ẋ = Ax+ Bu, with state x ∈ Rn and control u ∈ Rm. Several notions ofequivalence that are of practical importance. One may consider two systems equivalent if onecan be transformed into the other by coordinates changes, in state x̃ = Rx and control spaceũ = Su, and/or under feedback transformations ũ = u + Kx. In most applications of practicalinterest m < n. Hence B is a tall matrix with more rows than columns. Assuming no additionalconstraints on the controls u ∈ Rm, the behavior of the system is determined by the columnspaceof B, which, if B is of full rank, is an m-plane in n-space, i.e., a point on a Grassmannian.

The above examples have structure beyond just being sets of points. In order to be able to workwith concepts such as continuity and notions of derivatives one intuitively needs some notion ofdistance. Indeed, while one can start with even more general topological spaces, in the finitedimensional setting very little is lost if one requires that the set is equipped with at least somea-priori notion of distance. However, this basic notion of distance will be used primarily toensure that the topology of the manifold is reasonably nice. It should not be confused with theRiemannian metrics that will be introduced later, and which are closely related to curvature.

2.2 Some basic notions from metric spaces and topology

This subsection reviews basic definitions and properties of objects in metric spaces and topology.

Definition 2.1 A metric on a set X is a function d : X ×X 7→ R that satisfies

(i) For all x, y ∈ X, d(x, y) ≥ 0, and d(x, y) = 0 if and only if x = y (positive definiteness),

(ii) For all x, y ∈ X, d(x, y) = d(y, x) (symmetry), and

(iii) For all x, y, z ∈ X, d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).

A metric space is a pair (X, d) where X is a set and d is a metric of X.

A set X may be equipped with different metrics, which may or may not result in different notionsof continuity. For example, common metrics on Rn are

• the usual Euclidean distance defined by d2(x, y) = ‖y− x‖ =√

(y1 − x1)2 + . . . (yn − xn)2

• the taxi-cab metric defined by d1(x, y) =∑

i |yi − xi|, and

• the sup-norm d∞(x, y) = maxi |yi − xi|.


Definition 2.2 Suppose (X, d) is a metric space. A set O ⊆ X is called open if for every p ∈ Othere is an ε > 0 such that the (open) ε-ball Bp(ε) = {x ∈ X : d(x, p) < ε} is contained in O,i.e. Bp(ε) ⊆ O. A set F ⊆ X is called closed if X \ F is open.

Two metrics d and d′ on a space X are called equivalent if there exists constants c, C > 0 suchthat for all x, y ∈ X, cd′(x, y) ≤ d(x, y) ≤ Cd′(x, y).

Exercise 2.1 Show that the three metrics on Rn discussed above are equivalent.Show pictorially the meaning of above inequalities in terms of nested ε-balls with respect to thedifferent metrics. Conclude that the open sets in Rn are the same, independent of which of thesemetric employed to define open balls.

On any set X the discrete metric may be defined by d(x, y) = 1 is x 6= y and d(x, y) = 0 ifx = y. Given a metric d on a space X one may construct from it a bounded metric d̄ by settingd̄(x, y) = d(x, y) if d(x, y) ≤ 1 and d̄(x, y) = 1 else. Another useful bounded metric may beobtained by defining d̃ = d/(1 + d).

Exercise 2.2 Verify that the discrete and bounded metric described above are metrics.

Exercise 2.3 Let X be the set of all lines in the plane that pass through the origin. For lines`1 and `2 let d(`1, `2) to be the (smaller) angle between them. Verify that d is a metric on X.

Exercise 2.4 Show that if d is the discrete metric on a set X then every subset S ⊆ X is bothopen and closed.

Exercise 2.5 Consider a metric space (X, d) and the bounded metric d̄ (or d̃) constructed fromd as above. Show that a subset S ⊆ X is open in (X, d̄) if and only if it is open in (X, d).

The notions of open and closed do not require an underlying metric structure. The followingaxioms allow for a generalization to spaces without a metric:

Definition 2.3 A topology on a set X is a collection T ⊆ 2X of subsets of X that satisfies

(i) ∅ ∈ T and X ∈ T ,

(ii) T is closed under (arbitrary) unions, i.e., if {Oα : α ∈ A} ⊆ T then⋃α∈AOα ∈ T , and

(iii) T is closed under finite intersections, i.e., if Ok ∈ T , k = 1, 2, . . . n, then⋂nk=1Ok ∈ T .

A subset O ⊂ X is called open if O ∈ T . A subset F ⊂ X is called closed if X \ F ∈ T .A topological space is a pair (X, T ) where X is a set and T is a topology on X.

Note that an infinite intersection of open sets is not required to be open. The standard exampleis the real line with the usual topology and Ok = (− 1k ,

1k ). Clearly

⋂∞k=1Ok = {0} which is not

open (in the standard topology). An open neighborhood of p ∈ X is an open set which containsp. Technically a topological space is a pair (X, T ). But when the choice of the topology T isclear from the context it is customary to refer to X as a topological space. In many settingsit is convenient to work with a smaller set that generates the topology, and it often suffices toverify conditions for such basis elements to conclude that the conditions are actually satisfiedby all open sets.


Definition 2.4 A basis for a topology T on a set X is a collection B ⊆ T of open sets such thatfor every open set U ∈ T there exists a collection {Bα : α ∈ Λ} ⊆ B such that U =

⋃α∈ΛBα.

Exercise 2.6 Show that on a set X a collection B ⊆ 2X is a basis for a topology on X if andonly if B covers X (i.e., for every x ∈ X there exists B ∈ B such that x ∈ B) and for everyB1, B2 ∈ B, and for all x ∈ B1 ∩B2 there exists B3 ∈ B such that x ∈ B3 ⊆ B1 ∩B2.

Familiar examples of basis for topologies are the sets of all open intervals on the real line, andthe set of all open balls in Rn. Similarly, a collection S ⊆ 2X that covers X is called a subbasisfor the topology that consists of all finite intersections of all arbitrary unions of subcollectionsof sets in S. A typical application of this concept involves putting a topology on the tangentbundle of a manifold, on other bundles.

Definition 2.5 For a subset A ⊂ X of a topological space, the interior of A is the union of allopen subsets contained in A and it is denoted by Å or intA. The closure of A is the intersectionof all closed subsets containing A and it is denoted by A.

Exercise 2.7 Show that if Y ⊆ X is a subset of a topological space (X, T ) then the collectionT ′ = {O ∩ Y : O ∈ T } defines a topology on Y . This topology is called the subspace topology.

Exercise 2.8 Suppose that (X, d) is a metric space. Show that the collection of open sets (inthe sense of open in a metric space) defines a topology on X. This topology is called the metrictopology on (X, d).

Definition 2.6 A topological space is called second countable if it has a countable basis. A sub-set Z of a topological space X is called dense if every nonempty open subset of X intersects Z,and X is called separable if it has a countable dense subset.

One typically assumes that manifolds under investigation are second countable – this may bea consequence of such assumptions that they metric (or metrizable) spaces that are σ-compact(i.e. countable unions of compact sets, see below. Similarly, a topological space is called firstcountable if for every point x there exists a countable basis at x, that is a collection Bx of opensets each containing x such that for every open set U containing x, there exists a B ∈ Bx suchthat x ∈ B ⊆ U .

Example 2.1 Every metric space is first countable.Every separable metric space is second countable.Every Euclidean space Rn is second countable.

Exercise 2.9 Prove the assertions made in example 2.1.

Definition 2.7 Suppose X and Y (or, more precisely (X, TX) and (Y, TY )) are topologicalspaces. A map f : X 7→ Y is called continuous if for every open set O ⊆ Y the preimagef−1(O) ⊆ X is open (i.e. O ∈ TY =⇒ f−1(O) ∈ TX). (This is equivalent to f−1(F ) ⊆ X closedfor every F ⊆ Y closed.)A map f : X 7→ Y is called open if for every open set O ⊆ X the image f(O) ⊆ Y is open.A map f : X 7→ Y is called closed if for every closed set F ⊆ X the image f(F ) ⊆ Y is closed.


In the case that TX and TY are the metric topologies associated with metrics dX and dY on Xand Y , respectively, this notion of continuity agrees with the standard ε-δ characterization ofcontinuity. A function f : X 7→ Y between metric spaces is continuous (as defined above) if andonly if for every p ∈ X and for every ε > 0 there is a δ > 0 such that if q ∈ X with dX(p, q) < δthen dY (f(p), f(q)) < δ. Practically the notion of continuity captures the concept that smallchanges in the input of a function cause only small changes in the output.

Exercise 2.10 Consider the set R of real numbers with the usual topology T2, with the indiscretetopology T1 = {∅,R}, and with the discrete topology T3 in which every subset of R is open.For each pair (i, j) with i, j = 1, 2, 3 describe the set of continuous functions from (R, Ti) to(R, Tj). (Make a 3 × 3 table.) In particular, for which pairs is the identity function id : x 7→ xcontinuous? For which pair(s) are (only) the constant functions continuous, and for whichpair(s) are all functions continuous?

Exercise 2.11 Verify the assertion that in metric spaces the standard ε-δ characterization ofcontinuity agrees with the definition given above.


Definition 2.8 A map f : X 7→ Y between topological spaces is called a homeomorphism if

(i) f is a bijection, i.e. one-to-one and onto,

(ii) f is continuous, and

(iii) f−1 is continuous (i.e. f is open).

Two spaces topological spaces X and Y are called homeomorphic if there exists a homeomorphismf that maps X onto Y .

Do not confuse the term homeomorphism discussed here with homomorphism which refers tomaps that preserve algebraic relationships as in f(p · q) = f(p) · f(q).From a topological point of view homeomorphic spaces are considered as identical.

Exercise 2.12 Verify that the map f : (0, 1) 7→ R defined by f : x 7→ 1−2xx(x−1) is a homeomor-phism. (Calculate f ′ and consider lim

x→0+f(x) and lim

x→1−f(x).)

Exercise 2.13 Verify that the map f : R2 7→ B20(1) = {x ∈ R2 : ‖x‖ < 1} defined by f : x 7→x

1+‖x‖ is a homeomorphism.

While it seems intuitively clear that Rn and Rm are not homeomorphic for n 6= m, the proof(for general m and n) is surprisingly hard – usually utilizing tools from algebraic topology.Note that there exists for example a continuous function f : [0, 1] 7→ [0, 1] × [0, 1] that is con-tinuous and onto. However, the inverse f−1 can’t be continuous. For more details on theconstruction of such space-filling curves see e.g. Munkres, Topology, p. 271.

Definition 2.9 A subset A ⊆ X of a topologically space X is called connected if wheneverB,C ⊆ X are disjoint open sets such that B ∪C = A then A ⊆ B or A ⊆ C (i.e. A ∩B = ∅ orA ∩ C = ∅). If A is not connected then it is called disconnected.

Exercise 2.14 Suppose that f : X 7→ Y is a continuous map. Show that if f is onto and X isconnected then Y is connected.

Exercise 2.15 On a topological space X define the relation ∼ ⊆ X ×X by x ∼ y if there existsa connected subset C ⊆ X such that x ∈ C and y ∈ C. Show that ∼ is an equivalence relation.(The equivalence classes of this relation are called the (connected) components of the space X.)

In general topological spaces one works with a number of different notions of connectedness.Here only consider the following other notion, which is stronger than connectedness:

Definition 2.10 A subset A ⊆ X of a topologically space X is called path-connected if wheneverp, q ∈ A then there exist a continuous map γ : [0, 1] 7→ A such that γ(0) = p and γ(1) = q.

Exercise 2.16 Show that every path-connected set is connected.

Exercise 2.17 Show that the topologist’s sine curve, i.e. the set {(x, sin(

1x

)) ∈ R2 : x 6= 0} ∪

{0} × [−1, 1] is connected but not path-connected.


One of the most common uses of connectedness is the argument that if a function f : X 7→ R iscontinuous and locally constant then it is constant provided the domain X is connected. Here,locally constant means that every p ∈ X has an open neighborhood U (containing p) such thatthe restriction of f to U is constant.To clarify this argument, consider the function f : R \ {0} 7→ R defined by f : x 7→ 0 if x < 0and f : x 7→ 1 if x > 0. Clearly the derivative f ′ ≡ 0 vanishes identically, but f is not constant.Of course, the key is that the domain is not connected. Consequently, the vanishing of thederivative only assures that f is locally constant. It does not assure that f is constant.

Arguably the most important topological concept for basic analysis is compactness. One maythink of it as an outgrowth of the desire to generalize, or to get to the real basics of the importanttheorem that every continuous function f : [a, b] 7→ R defined on a closed bounded interval attainsits minimum and its maximum, i.e. there exist points x1, x2 ∈ [a, b] such that for all x ∈ [a, b],f(x1) ≤ f(x) ≤ f(x2). It is well-known from freshmen calculus that this assertion fails if eitherof the closedness or boundedness hypotheses is omitted. The closedness requirement naturallygeneralizes to general topological spaces, but the boundedness does not. For example, if Rnis equipped with the bounded metric d̄ = d21+d2 (where d2 is the standard Euclidean metric),then (Rn, d̄) still has the same topology as (Rn, d2) yet while K = Rn is closed and bounded in(Rn, d̄), it is not in (Rn, d2). Many different generalizations have been proposed to generalizethe basic idea of “closed and bounded” which is so useful in Rn(with its usual metric). Anyintroductory course in point-set topology will discuss such different notions of compactness. Itwas not until quite late into the 20th century that the following notion finally crystallized, andit became clear that it captures the fundamental features of the desired properties.

Definition 2.11 A subset K ⊆ X of a topological space X is called compact if every open coverof K has a finite subcover, i.e. if {Oα ⊆ X : α ∈ A} is a collection of open sets such that K ⊆⋃α∈AOα then there exists a finite subcollection {Oαj : j = 1, 2, . . . n} such that K ⊆

⋃nj=1Oαj

The Heine-Borel theorem asserts that every bounded closed interval in R is compact. Its proofmay be found in any advanced calculus text.

Exercise 2.18 Prove that if f : X 7→ Y is continuous and K ⊆ X is compact then the imagef(K) ⊆ Y is compact.In the case of Y = R this implies that there exist points p, q ∈ X at which f attains its globalminimum and global maximum, i.e. such that f(p) ≤ f(x) ≤ f(q) for all x ∈ X.

The corresponding property for preimages is frequently used to state that a map is radiallyunbounded as e.g. in the setting of Lyapunov stability of equilibria of dynamical systems.

Definition 2.12 A map f : X 7→ Y is called proper if for every compact set K ⊆ Y the preimagef−1(K) ⊆ X is compact.

Definition 2.13 A sequence {ak}k∈Z+ ⊆ X in a topological space X is said to converge if thereexists x ∈ X such that for every open set U ⊆ X containing x there exist a finite natural numberN such that an ∈ U for all n > N . Similarly, x is a limit point, or accumulation point of asubset A ⊆ X of for every set U ⊆ X containing x the set A ∩ U \ {x} is nonempty.

Exercise 2.19 Suppose A is an infinite subset of a compact space K. Show that A has alimit point in K. If, in addition, K is first countable then there exists a converging sequence{xk}k∈Z+.


The following separation axioms become useful when patching together local results, e.g. ob-tained in one coordinate chart at a time. This will be made precise when discussing partitionsof unity in a subsequent section.

Definition 2.14

• A topological space X is called a Hausdorff space if for every pair of distinct points p, q ∈ Xthere exist disjoint open sets U and V such that p ∈ U and q ∈ V .

• A Hausdorff space X is called completely regular if one-point sets are closed in X andif for every point p ∈ X and every closed set F ⊆ X not containing p there exists acontinuous function f : X 7→ R such that f(p) = 0 and f(x) = 1 for every x ∈ F .

• A Hausdorff space X is called normal if one-point sets are closed in X and if for everypair of disjoint closed sets F1, F2 ⊆ X there exists disjoint open sets U and V such thatF1 ⊆ U and F2 ⊆ V .

Finally the following notion of paracompactness is closely related to being normal and to metriz-ability, (indeed, metrizability is equivalent to paracompactness and local metrizability). Thusmany authors use paracompactness as a basic requirement when defining manifolds.

Definition 2.15 A subset S ⊆ X of a Hausdorff space X is called paracompact if every opencover of S has a locally finite open refinement. This means that if Uα, α ∈ A are open sets suchthat S ⊆

⋃α∈A Uα then there exist a collection of open sets Vβ, β ∈ B such that

• For every β ∈ B there exists an α ∈ A such that Vβ ⊆ Uα,

• S ⊆⋃β∈A Vβ, and

• every p ∈ S has an open neighborhood W which intersects only a finite number of the setsVβ, β ∈ B.


2.3 Local coordinate charts

This section defines the concept of a topological manifold which is to serve as a spring board forthe subsequent definition of a differentiable manifold. The main focus is on the concept of localcoordinate charts.

Definition 2.16 A topological manifold M is a metric space (M,d) such that for every p ∈Mthere exist n ∈ Z+, an open set U ⊆M containing p and a homeomorphism u : U 7→ Rn.

A few remarks:

• The metric d plays little role in the future – what is needed is really only a reasonablynice topological space. Metric, or more accurately, metrizable spaces just happen to haveabout the right set of properties needed throughout the standard development.

• The statement that U is homeomorphic to Rn may be replaced by U is homeomorphic toan open subset of Rn.

• If M is connected, then n is constant and is called the dimension of the manifold M , andM is called an m-manifold.

• The functions ui = xi◦u : U 7→ R are called local coordinates. Use xi : Rn 7→ R, i = 1, . . . nto denote the standard coordinate functions, defined by xi : (a1, . . . an) 7→ ai for everya = (a1, . . . an) ∈ Rn. The pair (u, U) is called a local coordinate chart about p. Note thatif (u, U) is a local coordinate chart about p then (ũ, U) defined by ũ : q 7→ u(q)− u(p) forq ∈ U is a local coordinate chart about p such that ũ(p) = 0. This is usually written asũ : (U, p) 7→ (Rn, 0).

If (u, U) and v, V are coordinate charts about p ∈M (i.e. in particular p ∈ U ∩ V , then

v ◦ u−1 : u(U ∩ V ) 7→ v(U ∩ V ) (8)

is a homeomorphism between open subsets of Rn.

Continue with a short list of examples of manifolds and coordinate charts.

• For any n ≥ 0 the n-dimensional Euclidean space Rn is an n-dimensional manifold withthe single coordinate chart (id,Rn).


• The open ball Bp(r) = {x ∈ Rn : ‖x−p‖ < r} of radius r about p ∈ Rn is an n-dimensionalmanifold with a single chart given by U = Bp(r) and u(x) = (x− p)/(r − ‖x− p‖).

• Every open subset of an n-dimensional manifold is itself an n-dimensional manifold.

• Identify the space Mm,n(R) of m-by-n matrices with real entries with the space Rmn.E.g. in the 2× 2-case simply identify

u :(a bc d

)7→

abcd

With this identification Mm,n(R) inherits a natural metric structure. Clearly this showsthat the entire space Mm,n(R) is an nm-dimensional manifold.Of more interest are various subspaces of Mm,n(R). Typical examples are the general lineargroup GL(n,R) = {A ∈ Mn,n(R) : detA 6= 0} and its subset of orientation preservingnonsingular matrices GL+(n,R) = {A ∈ Mn,n(R) : detA > 0}. Both are n2-dimensionalmanifolds. The argument uses that det is a polynomial function in the entries of the matrix,and hence it is continuous. Consequently, the preimages det−1(R \ {0}) and det−1(0,∞))of open sets are open subsets of Mn,n(R).

Further examples are the special linear groups SL(n,R) = {A ∈Mn,n(R) : detA = 1}, theorthogonal groups O(n) = {A ∈ Mn,n(R) : ATA = I}, and the special orthogonal groupsSO(n) = O(n)∩SL(n,R). Unlike the previous examples – which are open subsets – thesemanifolds are defined by closed conditions (i.e. “=” as opposed to “6=”, “”).These examples will be revisited later when tools from differential calculus on manifoldswill make it easy to establish when such subsets defined by closed conditions give rise tomanifolds.

• The m-sphere Sm(r) = {x ∈ Rm+1 : ‖x‖ = r} is a m-manifold. The case m = 0 is specialwith S0(r) = {−r, r} ⊆ R consisting of only two points, and thus being disconnected. The1-sphere is a circle, and one needs at least two charts, e.g. U = S1(r) \ {(−r, 0)} andV = S1(r) \ {(r, 0)}. Use u(x, y) = atan2 (x, y) taking values in (−π, π) and v(x, y) =atan2 (x, y) taking values in (0, 2π). Note that the local coordinates agree essentially withthe angle of polar coordinates.

Exercise 2.20 Construct a collection of charts for the 2-sphere S2(r) by explicitly adapt-ing the spherical coordinates u(x, y, z) = (θ(x, y, z), φ(x, y, z)) ∈ (0, 2π)× (0, π) to differentsubsets resulting from different “cuts” What is the minimal number of cuts (minimal num-ber of charts) needed to cover S2?

A different set of coordinates, that is particularly useful for higher dimensional spheres isbased on stereographic projections: Consider the two subsets U± = Sm(r)\{(0, 0, . . . ,±r)}and define the maps u± : U± 7→ Rm by

u±(x) =2r

r ∓ xm+1(x1, . . . xm) (9)

Graphically (u±(x),∓r) ∈ Rm+1) is the point where the hyperplane xm+1 = ∓r intersectsthe line that passes through the point x ∈ Sm(r) and through (0, 0, . . . ,±r).


Exercise 2.21 For the inverse maps u−1± : Rm 7→ U± ⊆ Sm(r), derive the formulae

x = u−1± (y) =

(2r

1 +∥∥ y

2r

∥∥2 (y12r) , . . . 2r1 + ∥∥ y2r∥∥2(ym

2r

),∓ r ·

1−∥∥ y

2r

∥∥21 +

∥∥ y2r

∥∥2)

(10)

Use this to obtain explicit formulae for the “transition maps” u∓ ◦ u−1± : Rm 7→ Rm.What do these maps do graphically – e.g. which sets do they leave fixed?What are the images of (special) lines and circles?

• If Mm and Nn are m- and n−dimensional manifolds, respectively, then the Cartesianproduct M ×N is an (m+ n)-dimensional manifold: Suppose (p, q) ∈M ×N and (u, U)and (v, V ) are coordinate charts about p and q, then (u× v, U × V ) is a coordinate chartabout (p, q) where (u, v)(a, b) = (u(a), v(b)) ∈ Rm × Rn.A typical example uses that the circle S1 = {x ∈ R2 : ‖x‖ = 1} is a manifold to establishthat the torus T 2 = S1 × S1 is a 2-dimensional manifold.

• Next construct coordinate charts for real projective spaces. On Rm+1 \ {0} define theequivalence relation x ∼ y if x = λy for some λ ∈ R. The m-dimensional real projectivespace is defined as the quotient IPm =

(Rm+1 \ {0}

)/ ∼, i.e. a point [x] ∈ IPm is the

equivalence class [x] = {y ∈ Rm+1 \ {0} : y ∼ x}. Intuitively think of IPm as the space ofall lines in Rm+1 that pass through the origin, or as the m-sphere Sm with antipodal pointsx and −x identified. More graphically, one may obtain IP2 by sewing a disk to the (onlyone!) edge of a Möbius strip. For j = 1, . . .m consider the sets Uj = {[x] ∈ IPm : xj 6= 0}and coordinates maps (homogeneous coordinates)

uj([x]) =(x1xj, . . .

xj−1xj

,xj+1xj

, . . .xm+1xj

)(11)

Clearly the value of uj([x]) does not depend on the choice of the representative x ∈ [x].The inverse is given by u−1j (y1, . . . ym) = [y1, . . . , yi−1, 1, yi+1, . . . ym].

A complete discussion of these coordinate maps (they are supposed to be homeomorphisms)is straightforward but technical, in terms of the quotient topology. For an introductorydiscussion of quotient maps, and quotient topologies see e.g. Munkres (Topology, a firstcourse”, p.134). A main issue is to assure that the quotient topology is not pathological.Just as a reference, a surjective map f : X 7→ Y is called a quotient map if O ⊂ Y is openif and only if f−1(O) ⊆ X is open. For any map f : X 7→ A from a topological space X toa set A there is exactly one topology on A, called the quotient topology, such that f is aquotient map. In the case that A is a set of equivalence classes on X, A with this topologyis called a quotient space of X.

• Two-dimensional surfaces in R3 , or more generally m-dimensional hypersurfaces in Rm+1are familiar manifolds. The graph {(x, f(x)) : x ∈ Rm} ⊆ Rm+1 of a continuous functionf : Rm 7→ R is a manifold with the single chart u : (x, f(x)) 7→ x ∈ Rm.

More interesting are hypersurfaces that arise as preimages (“zero-sets”)M = F−1({0}) ={x ∈Rm+1 : F (x) = 0} of functions F : Rm+1 7→ R, or that are given by parameterizationsF : Rm 7→ Rm+1 and M = F (Rm).More generally, if M and N are a manifolds and Φ: M 7→ N then Φ(M) ⊆ N may be


a manifold. Similarly, if P ⊆ N is a submanifold, then Φ−1(P ) ⊆ M might be a sub-manifold of M . To avoid unnecessary duplication and difficulties we shall discuss theseconstructions only in the setting of differentiable manifolds, in a subsequent section. Hereonly briefly consider two examples: Let f, g : R2 7→ R be defined by f : (x, y) 7→ x2 +y2−1and g : (x, y) 7→ xy. Then f−1(0) is the 1-sphere, while g−1(0) is not a manifold. Thestandard criterion that distinguishes these examples relies on derivatives: While (Df)never vanishes where f vanishes, (Dg) and g have common zeros – which are potentiallytroublesome points.

• There is a very rich world of complex manifolds – but we will not have the opportunity toexplore this in any depth in this course. Within the context of this section – coordinatecharts for topological manifolds – complex manifolds do not offer any new features. Butin the framework of differentiable manifolds, the much more rigid structure of complexdifferentiability opens completely new issues, far beyond this course.


2.4 Differentiation: Notation and review

This section fixes some notation for partial derivatives of maps between Euclidean spaces, andcontrasts this with a coordinate-free description of differentiation.Let ei = (0, 0, . . . 0, 1, 0 . . . 0)T ∈ Rn denote the standard i-th basis vector. For a functionf : U 7→ R defined on an open subset U ⊆ Rn and a ∈ U the i-the partial derivative of f at a isdefined (and denoted) by

(Dif)(a) = limh→0

1h (f(a+ hei)− f(a)) (12)

provided the limit exists. To denote different degrees of regularity, use the following notation:f ∈ C0(U) f is continuous on U (i.e., f is continuous at every p ∈ U)f ∈ Cr(U) all partial derivatives of up to order r of f exist and are continuous on U ,f ∈ C∞(U) all partial derivatives of all orders of f exist and are continuous on U ,f ∈ Cω(U) f is real analytic on U (f ∈ C∞(U) and f agrees (locally) with its Taylor series).

For a function f : A 7→ R defined on a set A ⊆ Rm one says f ∈ Cα(A) if there exist an openset O ⊆ Rm such that A ⊆ O and an extension f̄ of f to O (i.e. f̄ |A = f), and f̄ ∈ C

α(O).

If f : A ⊆ Rn 7→ Rm, then f ∈ Cα(A,Rm) if each coordinate function fk = xk ◦ f ∈ Cα(A).If f ∈ C1(Rn,Rm) then the Jacobian matrix is

(Df) = (Djf i) i = 1, . . .mj = 1, . . . n

=

D1f1 . . . Dnf

1

.... . .

...D1f

m . . . Dnfm

(13)For convenience identify the space Mm,n(R) of real m × n matrices with Rmn. Thus, if f ∈Cr(Rn,Rm), then (Df) ∈ Cr−1(Rn,Rmn), and (Dkf) ∈ Cr−k(Rn,Rmnk).The chain rule asserts that if U ⊆ Rm and V ⊆ Rn are open sets, g ∈ Cr(U,Rn) and f ∈Cr(V,Rp) with r ≥ 1, g(U) ⊆ V and a ∈ U , then f ◦ g is differentiable at a and D(f ◦ g)(a) =(Df)(g(a)) · (Dg)(a) (matrix-multiplication).

Theorem 2.1 (Implicit function theorem) Suppose U ⊆ Rm+n is open, (a, b) ∈ U and f ∈Cr(U,Rm) with r ≥ 1, and f(a, b) = 0. If the matrix of partial derivatives

(Dn+jf

i(a, b))i,j=1...m

is nonsingular then there exist open sets V ⊆ Rn and W ⊆ Rm with a ∈ V and b ∈ W and aunique function g : V 7→W such that f(x, g(x)) = 0 for all x ∈ V . Moreover, g ∈ Cr(V,W ).

Differentiability (as opposed to mere existence of partial derivatives) may be described in acoordinate-free way. Consider finite dimensional normed linear spaces V,W and let U ⊆ V beopen. Recall, a norm is a map ‖ · ‖ : V 7→ R such that ‖v‖ ≥ 0 for all v ∈ V , ‖v‖ = 0 if and onlyof v = 0, ‖λv‖ = |λ|‖v‖ for all v ∈ V and λ ∈ R, and ‖v + w‖ ≤ ‖v‖+ ‖w‖ for all v, w ∈ V . Asfinite dimensional linear spaces V and W are isomorphic to some Euclidean spaces Rn and Rm,but here the objective is to not fix any bases.

Definition 2.17 A map f : U 7→ W is differentiable at a ∈ U if there exists a linear mapL = La,f ∈ Hom(V,W ) such that f(a + h) = f(a) + L(h) + o(‖h‖). (This means that thereexists a map η : U 7→ W (depending on a and f) such that f(a + h) = f(a) + L(h) + ‖h‖η(h)and ‖η(h)‖ −→ 0 at ‖h‖ −→ 0.)


Exercise 2.22 Show that if f is differentiable at a then the linear map of the preceding definitionis uniquely determined. (Suppose there are two such linear maps. Show that their differencesatisfies L(h) −M(h) = o(‖h‖).) Therefore it is justified to talk about the derivative of f at a– and use the notation f ′(a).

A map f is called differentiable on an open set U if f is differentiable at all a ∈ U .

Note that if f : U ⊆ V 7→W then f ′ : U 7→ Hom(V,W ). But the space Hom(V,W ) of linear mapsfrom V to W is itself a linear space (finite dimensional, normed – with the induced norm ‖L‖ =max{‖Lv‖ : ‖v‖ = 1}). Hence one may naturally define higher order derivatives as linear mapsf ′′ : U 7→ Hom(V,Hom(V,W )) ∼= Hom(V ⊗ V,W ) and inductively f (k) : U 7→ Hom(

⊗ki=1 V,W ).

For comparison, if f : Rn 7→ Rm then (Df) is an (m×n) matrix, and, naively, (D2f) is some sortof (m × n × n) object which takes three inputs, a point where it is evaluated and two vectors,and its output is an m-vector. . .To summarize a few basic properties

(i) If f is differentiable then f is continuous.

(ii) If f and g are differentiable and λ ∈ R then (f + g)′ = f ′ + g′ and (λf)′ = λf ′.

(iii) If f is constant then f ′ ≡ 0. The converse is true if U is (path)connected.

(iv) If L ∈ Hom(V,W ), b ∈W and f = L+ b (i.e. f : v 7→ L(v) + b) then f ′ = L.

(v) (Chain-rule). Let V1, V2, V3 be normed linear spaces. Suppose U1 ⊆ V1 and U2 ⊆ V2 areopen, g : U1 7→ V2, f : U2 7→ V3, and g(U1) ⊆ U2. If f and g are differentiable, then so isf ◦ g and (f ◦ g)′ = (f ′ ◦ g) · g′.This notation is to be interpreted as follows: For p ∈ U1 and v ∈ V1, g′(p) ∈ Hom(V1, V2)and hence g′(p)(v) ∈ V2. Similarly, f ′(g(p)) ∈ Hom(V2, V3). Therefore f ′(g(p))(g′(p)(v)) ∈V3. This matches with (f ◦ g)′(p) ∈ Hom(V1, V3) and hence (f ◦ g)′(p)(v) ∈ V3.

2.5 Differentiable structures

In general, manifolds do not have a linear, not even an additive structure. Thus expressionsreminiscent of f ′(x) = lim

h→01h (f(x+ h)− f(x)) are meaningless for maps f : M 7→ N between

manifolds (unless one considers points in M as distributions on the algebra of smooth functions).A natural way to define a notion of differentiability on manifolds is to utilize coordinate charts(u, U) on M and (v, V ) on N to relate f : U 7→ V to the map v ◦ f ◦ u−1 between Euclideanspaces. The main concern is to ensure that any so-defined notion of differentiation on a manifolddoes not depend on the particular choice of coordinates. This leads naturally to the concept ofdifferentiable structures.

Definition 2.18 Two charts (u1, U1) and (u2, U2) on a manifold M are Cr-related (r = 1, 2, . . . ,∞, ω) if the maps u2 ◦u−11 and u1 ◦u

−12 are C

r-maps as maps between Euclidean spaces on theirrespective domains u1(U1 ∩ U2) and u2(U1 ∩ U2).

Definition 2.19 A Cr-differentiable structure on a manifold M is a maximal atlas D, thatis, a maximal collection of coordinate charts that covers M and which is such that any twocharts (u1, U1), (u2, U2) ∈ D are Cr-related. A manifold M together with a Cr-differentiablestructure D is called a Cr-manifold, or simply, a differentiable manifold (if r is understood,usually r =∞).


Definition 2.20 Two Cr-manifolds (M,D) and (N,D′) are called diffeomorphic if there existsa bijection Φ: M 7→ N such that (v, V ) ∈ D′ if and only if (v ◦ Φ,Φ−1(V )) ∈ D.The map Φ is called a diffeomorphism.

It is easy to see that every diffeomorphism must be continuous. Since the inverse Φ−1 is auto-matically a diffeomorphism, Φ is automatically a homeomorphism. However, manifolds may behomeomorphic without being diffeomorphic, see below.

Proposition 2.2 Every Cr-atlas is contained in a unique Cr-differentiable structure.

Proof. Let U = {(uα, Uα) : α ∈ A} be a Cr-atlas for a manifold M – i.e. any two charts(uα, Uα), (uβ, Uβ) ∈ U are Cr-related and M ⊆

⋃α∈A Uα. Define D to be the collection of all

coordinate charts (vα, Vα) on M which (each) are Cr-related to every (uα, Uα) ∈ U .Maximality of D is clear. To verify that any two charts (vα, Vα), (vβ, Vβ) ∈ D are Cr-relatedit suffices to show that vβ ◦ v−1α : vα(Vα ∩ Vβ) 7→ Rm is locally Cr. Thus suppose that x ∈vα(Vα ∩ Vβ) ⊆ Rm and let p = v−1α (x) ∈ M . Since U is an atlas of M , it contains a chart(u, U) ∈ U about p. On the set vα(U ∩ Vα ∩ Vβ) one may write

vβ ◦ v−1α =(vβ ◦ u−1

)◦(u ◦ v−1α

)as a composition of Cr-maps – and hence vβ ◦ v−1α is locally Cr on vα(U ∩ Vα ∩ Vβ). Since thelatter set is open in Rm, vβ ◦ v−1α is Cr.Regarding uniqueness, suppose D′ is any Cr differentiable structure containing U . Then bydefinition every (v, V ) ∈ D′ is Cr-related to every (u, U) ∈ U , and consequently (v, V ) ∈ D, i.e.D′ ⊆ D. Since a differentiable structure is maximal by definition, also D ⊆ D′, i.e. D = D′.

Definition 2.21 Let (M,D) and (N,D′) be differentiable manifolds of class Cr and Cs, respec-tively, and k ≤ min{r, s}. A map Φ: M 7→ N is called differentiable of class Ck if for any charts(u, U) ∈ D and (v, V ) ∈ D′ the map v ◦ Φ ◦ u−1 is of class Ck on its domain.

• If M = Rm and N = Rn (with the usual differentiable structures) then Φ: M 7→ N isdifferentiable if it is differentiable in the usual sense.

• A map Φ: M 7→ Rn is differentiable if and only if each coordinate function Φk = xk ◦Φ isdifferentiable.

• Coordinate maps u are diffeomorphisms from U to u(U).

• A map Φ: M 7→ Rn is a diffeomorphism if and only if it is bijective, differentiable, and itsinverse is differentiable.

Proposition 2.3 Every differentiable map Φ: M 7→ N between manifolds is continuous.

Proof. W ⊆ N be open and suppose p ∈ Φ−1(W ) ⊆ M . Choose charts (u, U) about p ∈ Mand (v, V ) about Φ(p) ∈ N . Then (V ∩W ) ⊆ N is an open set containing Φ(p), and hencev(V ∩W ) ⊆ Rn is an open set containing (v ◦Φ)(p). Write p = (u−1 ◦ (v ◦Φ ◦u−1)−1)(v(Φ(p))).Then p is contained in the set u−1((v ◦ Φ ◦ u−1)−1(v(V ∩ W ))) which is an open subset ofU ∩ |phi−1(W ) containing p since u−1 and v are homeomorphisms, and (v ◦Φ ◦ u−1) as a differ-entiable map between Euclidean spaces is continuous. This shows that Φ−1(W ) ⊆ M is open,and hence Φ is continuous.


Exercise 2.23 Show that the identity map idM : M 7→M on a Cr-manifold M is a Cr-map.Suppose that M, N, and P are Cr-manifolds with A ⊆M and B ⊆ N . Show that if g ∈ Cr(A,N)and f ∈ Cr(B,P ) then f ◦ g ∈ Cr(A ∩ g−1(B), P ).

Exercise 2.24 Consider M = R with charts (uk, U), k = 1, 2, 3 where U = R and u1(x) = x3,u2(x) = x, and u3(x) = x1/3. In each case provide an example of another chart (vk, Vk) that iscontained in the unique C1-differentiable structure Dk on M that contains (uk, Uk).Explain why Dk are (pairwise) different, give examples of charts contained in their intersections(or explain why the intersections are empty). Demonstrate that (M,Dk) are diffeomorphic.

Exercise 2.25 (Continuation of exercise 2.24). Consider the maps Φi,j,k : (M,Di) 7→ (M,Dj)defined by Φi,j,k(x) = xk for k = 19 ,

13 , 1 , 3 , 9. Which of these maps are differentiable? Which

maps are diffeomorphisms?

Note that every differentiable structure D of class Cr, r ≥ 1 may be regarded as an atlas of classCr−1, and hence is contained in a unique differentiable structure D′ of class Cr−1. Consequentlythe class of a manifold can be lowered at will, by adding new charts to an atlas. More importantis that every Cr-differentiable structure with r ≥ 1 contains a C∞ differentiable structure (seealso below). Consequently one routinely restricts one’s attention to C∞-manifolds.

Blanket hypothesis. Unless otherwise stated, all manifolds and maps consideredhenceforth are assumed to be C∞ manifolds and C∞-maps, respectively.

A few side-remarks:

A (very hard) theorem by Whitney asserts that every C1 differentiable structurecontains a Cω structure. On the other hand, Kervaire has given an example of a 10dimensional C0 manifold which admits no differentiable structure (hard!).The spheres Sn have unique differentiable structures for n ≤ 6, but Milnor showedin 1958 that S7 admits 28 non-diffeomorphic differentiable structures! He gave anexplicit construction of such exotic sphere that basically glues together two copies ofB3 × S3 along their boundaries S3 × S3 but identifying the point (a, b) of one copywith the point a, a2ba−1) of the other. (This uses a group structure closely relatedto quaternions and SO(3).) However, for any n there is only a finite number of“diffeo-classes” on Sn.For n 6= 4 the Euclidean space Rn has a unique differentiable structure, but R4 hasat least 3 non-diffeomorphic differentiable structures (1982).

This material makes for great reading, and there continue to be new exciting resultsdiscovered on a regular basis. It is easy to find exciting discussions of this materialon-line, incl. the connections to the Poincaré conjecture and Perelman’s proof of thelatter – which goes far beyond topology and uses geometry in essential ways.)

Some of this material is well suited for a survey paper (such as in a minor thesis,and for presentations in class or in graduate seminars.


2.6 Differentiating differentiable maps

This section starts by defining (partial) derivatives of maps on coordinate charts. This notionwill enable us to define submersions, immersions, and imbeddings. These in turn are usefultools to construct new differentiable manifolds from known ones. As examples, hypersurfacesand some matrix submanifolds are revisited, and it is shown that every compact manifold canbe imbedded into a Euclidean space.

Recall the definition of the i-th partial derivative of a function f : Rn 7→ R at a point a ∈ Rn:Dif(a) = lim

h→01h (f(a1, . . . , ai−1, ai + h, ai−1, . . . , an)− f(a)).

The chronological notation aDif employed in geometric control will not be used in the sequel.

Definition 2.22 Suppose M is a differentiable manifold and f : M 7→ R. The i-th partialderivatives of f in a local coordinate chart (u, U) at a point p ∈ U is defined as

∂f

∂ui

∣∣∣∣p

= Di(f ◦ u−1)∣∣u(p)

(14)

Note, if γ : (−ε, ε) 7→ U ⊆ M is a curve such that γ(0) = p, ui(γ(t)) = t and uj(γ(t)) = uj(p)for j 6= i then for f : U 7→ R

∂f

∂ui

∣∣∣∣p

= Di(f ◦ u−1)∣∣u(γ(0))

= limh→0

1h (f(γ(h))− f(p)) (15)

This curve γ is adapted in a special way to the specific local coordinate chart. The next sectionwill generalize this setting along its way to construct the tangent bundle.

• If f = uj then ∂uj∂ui

= δi,j (Kronecker delta, i.e., δi,j = 1 if i = j and δi,j = 0 else).

• If M = Rn with the identity chart then ∂f∂ui

= Dif .

Proposition 2.4 (Chain-rule) Let (u, U) and (v, V ) be local coordinate charts, p ∈ U ∩V andf : M 7→ R. Then

∂f

∂vj

∣∣∣∣p

=n∑i=1

∂f

∂ui

∣∣∣∣p

∂ui

∂vj

∣∣∣∣p

(16)

In this case, the relation uses only two indices and may be nicely be written in matrix from

(∂f

∂v1, . . . ,

∂f

∂vn

)=(∂f

∂u1, . . . ,

∂f

∂un

)∂u1

∂v1. . . ∂u

1

∂vn...

. . ....

∂un

∂v1. . . ∂u

n

∂vn

. (17)However, many subsequent formulas will involve more indices and hence generally will not beamenable to simple matrix thinking. Instead, most helpful is a careful and consistent choiceof indices written as subscripts and superscripts. This goes as far as the “Einstein summationconvention” (not used in these notes), which omits the summation signs and instead tacitlyimplies that if an index occurs once as a lower and once as an upper index, then it is understoodthat this index is to be summed over. Yet even without using that summation convention, it ishelpful to carefully check balanced occurrences as upper and lower indices.


Proof. Use the coordinate maps and the chain-rule in Euclidean spaces

∂f∂vj

∣∣∣p

= Dj(f ◦ v−1)∣∣v(p)

= Dj((f ◦ u−1) ◦ (u ◦ v−1)

)∣∣v(p)

=n∑i=1

Di((f ◦ u−1)

∣∣(u◦v−1)(v(p)) · Dk(u

i ◦ v−1))∣∣∣v(p)

=n∑i=1

∂f∂ui

∣∣∣p

∂ui

∂vj

∣∣∣p.

(18)

Proposition 2.5 (Product-rule) Suppose (u, U) is a chart, p ∈ U and f, g : M 7→ R. Then

∂(fg)∂vj

∣∣∣∣p

=∂f

∂vj

∣∣∣∣p

· g(p) + f(p) · ∂g∂vj

∣∣∣∣p

(19)

Exercise 2.26 Prove the product rule for functions on a manifold as stated above.

Proposition 2.6 Let (u, U) and (v, V ) be local coordinate charts, about p ∈ U ∩ V . Then theJacobian matrix (

∂ui

∂vj

∣∣∣∣p

)i = 1, . . . nj = 1, . . . n

is invertible. (20)

Proof. The chain-rule gives δi,k = ∂ui

∂uk

∣∣∣p

=n∑j=1

∂ui

∂vj

∣∣∣p· ∂vj∂uk

∣∣∣p. Hence the Jacobian matrices

A =

(∂ui

∂vj

∣∣∣∣p

)i = 1, . . . nj = 1, . . . n

and B =

(∂vj

∂uk

∣∣∣∣p

)j = 1, . . . nk = 1, . . . n

(21)

satisfy A ·B = I, and similarly B ·A = I, i.e A is invertible.

Proposition 2.7 Let Φ: Mm 7→ Nn be a differentiable map between manifolds. If (u, U) and(ū, Ū) are local coordinate charts about p ∈M , and (v, V ) and (v̄, V̄ ) are local coordinate chartsabout Φ(p) ∈ N , then the Jacobian matrices (at p ∈ U ∩ Ū)(

∂(vi ◦ Φ)∂uj

∣∣∣∣p

)i = 1, . . . nj = 1, . . . n

and

(∂(v̄i ◦ Φ)∂ūj

∣∣∣∣p

)i = 1, . . . nj = 1, . . . n

have the same rank (22)

Proof. Applying the chain-rule twice yields

∂(v̄i◦Φ)∂ūj

∣∣∣p

= Dj(v̄i ◦ Φ ◦ ū−1

)∣∣ū(p)

= Dj((v̄i ◦ v−1

)◦(v ◦ Φ ◦ u−1

)◦(u ◦ ū−1

))∣∣ū(p)

=n∑k=1

m∑̀=1

Dk(v̄i ◦ v−1

)∣∣(v◦Φ)(p) · D`

(vk ◦ Φ ◦ u−1

)∣∣u(p)· Dj

(u` ◦ ū−1

)∣∣ū(p)

=n∑k=1

m∑̀=1

∂v̄i

∂vk

∣∣∣Φ(p)

∂(vk◦Φ)∂u`

∣∣∣p

∂u`

∂ūj

∣∣∣p

(23)


The proposition follows from the invertibility of the Jacobian matrices(∂v̄i

∂vk

∣∣∣∣Φ(p)

)i = 1, . . . nk = 1, . . . n

and

(∂u`

∂ūj

∣∣∣∣p

)` = 1, . . .mj = 1, . . .m

. (24)

Definition 2.23 Let Φ: Mm 7→ Nn be a differentiable map between manifolds. The rank of Φat p ∈M is defined as the rank of the Jacobian matrix(

∂(vi ◦ Φ)∂uj

∣∣∣∣p

)i = 1, . . . nj = 1, . . .m

(25)

where (u, U) and (v, V ) are any local coordinate charts about p ∈M and Φ(p) ∈ N , respectively.

This definition is justified – the rank is well-defined – by virtue of the preceding proposition:The rank of Φ at p (written rankpΦ) is independent of the choice of local coordinates employed.

Definition 2.24 Suppose Φ: M 7→ N is a differentiable map between manifolds.

• p ∈M is called a regular point of Φ if rankpΦ = dimN .

• p ∈M is called a critical point of Φ if rankpΦ < dimN .

• q ∈ N is called a regular value of Φ if every point p ∈ Φ−1(q) ⊆M is a regular point of Φ.

• q ∈ N is called a critical value of Φ if q = Φ(p) for some critical point p of Φ.

Note that every point q ∈ N \ Φ(M) is (vacuously) a regular value of Φ.Intuitively one expects that the set of critical values is a small subset of N . To make this preciseit suffices to have a notion of sets of measure zero on the manifold. (A notion of measure zerodoes not require a notion of a measure on the manifold!) Recall that a subset A ⊆ Rn hasmeasure zero, if for every ε > 0 there exist a countable collection of open cubes

Ci(pi, ri) = {q ∈ Rn : ‖q − p‖∞ < ri} such that A ⊆∞⋃i=1

Ci(pi, ri) and∞∑i=1

rni < ε. (26)

Definition 2.25 A subset S ⊆Mn of a manifold M has measure zero if there exist a sequenceof charts (ui, U

Classnotes for APM 581 Geometry and Control of Dynamical …kawski/classes/apm581/10fall/... ·...

Documents

Transcript of Classnotes for APM 581 Geometry and Control of Dynamical …kawski/classes/apm581/10fall/... ·...