Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

The World Is Not Flat:An Introduction to Modern Geometry

Richard G. Ligo

The University of Iowa

September 15, 2015

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

The story of a hunting party

What color was the bear?

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

The story of a hunting party

What color was the bear?

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

The story of a hunting party

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Overview

I Introduction

I Euclid’s foundations

I Gauss and all his friends

I Riemann’s revolution

I Conclusion

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Euclid of Alexandria

I Active around 300 B.C.

I Authored several mathematical texts

I Transformed geometry into a rigorous discipline

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Euclid’s Elements

The five axioms:

1. A straight line segment can be drawn joining any twopoints.

2. Any straight line segment can be extended indefinitelyin a straight line.

3. Given any straight line segment, a circle can be drawnhaving the segment as radius and one endpoint ascenter.

4. All right angles are congruent.

5. Given any straight line and a point off the given line,there exists exactly one line through the point notintersecting the given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

The fathers of non-Euclidean Geometry

I Carl Frederich Gauss (1777-1855)

I Nikolai Lobachevsky (1792-1856)

I Janos Bolyai (1802-1860)

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Given any straight line and a point off the given line,

Euclidean: there exists exactly one line through the pointnot intersecting the given line.

Elliptic: there exist no lines through the point notintersecting the given line.

Hyperbolic: there exist at least two lines through the pointnot intersecting the given line.

“I created a new, different world out of nothing.”- Bolyai, in a letter to his father

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Straight lines

What makes a curve in a surface a “straight line”?

I It will (at least locally) represent the shortest distancebetween two points.

I It will have zero acceleration in the surface.

The generalization of a straight line in a non-Euclideanspace is called a geodesic.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Straight lines

What makes a curve in a surface a “straight line”?

I It will (at least locally) represent the shortest distancebetween two points.

I It will have zero acceleration in the surface.

The generalization of a straight line in a non-Euclideanspace is called a geodesic.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Straight lines

What makes a curve in a surface a “straight line”?

I It will (at least locally) represent the shortest distancebetween two points.

I It will have zero acceleration in the surface.

The generalization of a straight line in a non-Euclideanspace is called a geodesic.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Euclidean: Given any straight line and a point off the givenline, there exists exactly one line through the point notintersecting the given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Euclidean: Given any straight line and a point off the givenline, there exists exactly one line through the point notintersecting the given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Euclidean: Given any straight line and a point off the givenline, there exists exactly one line through the point notintersecting the given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Elliptic: Given any straight line and a point off the givenline, there exists no lines through the point not intersectingthe given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Elliptic: Given any straight line and a point off the givenline, there exists no lines through the point not intersectingthe given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Elliptic: Given any straight line and a point off the givenline, there exists no lines through the point not intersectingthe given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Hyperbolic: Given any straight line and a point off the givenline, there exists at least two lines through the point notintersecting the given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Hyperbolic: Given any straight line and a point off the givenline, there exists at least two lines through the point notintersecting the given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

A family of parallel postulates

Hyperbolic: Given any straight line and a point off the givenline, there exists at least two lines through the point notintersecting the given line.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Curve curvature

How can we formally describe curvature?

To calculate the curvature of a parametric curve, we find itssecond derivative.

Curvature is typically represented by the Greek letter κ.

zero κ low κ high κ

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Curve curvature

How can we formally describe curvature?

To calculate the curvature of a parametric curve, we find itssecond derivative.

Curvature is typically represented by the Greek letter κ.

zero κ low κ high κ

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Curve curvature

How can we formally describe curvature?

To calculate the curvature of a parametric curve, we find itssecond derivative.

Curvature is typically represented by the Greek letter κ.

zero κ low κ high κ

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Curve curvature

Circles have uniform curvature; specifically, κ = 1r .

κ = 12 κ = 1 κ = 3

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Curve curvature

In some sense, the curvature of a curve at a point is thecurvature of the largest circle one can make tangent to thecurve at that point.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

There are several ways to determine the curvature at a pointon a surface, but the following process is the most intuitive:

1. Construct the tangent plane to the surface at the point.

2. Consider every plane perpendicular to the tangent plane.

3. Each of these perpendicular planes will intersect thesurface in some curve.

4. Select the two curves with the “most different”curvatures at the point. These are called the principalcurvatures.

5. Multiply their curvatures and choose the appropriatesign to obtain the surface’s curvature at that point.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

K (p) = 0 · 1r = 0

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

K (p) = 1r ·

1r = 1

r2> 0

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

R

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

K (p) = −k1 · k2 < 0

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Gauss curvature

Thus, we can separate curvature at a point on a surface intothree cases:

Zero: the surface can be perfectly flattened.

Positive: the surface will rip when flattened.

Negative: the surface will fold when flattened.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

The Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem illustrates a connection betweentopological and geometric properties.

Theorem: If M is a compact, two-dimensional, Riemannianmanifold with boundary ∂M, then∫

MK dA +

∫∂M

kg ds = 2πχ(M),

Where K is the Gauss curvature on M, kg is the geodesiccurvature on ∂M, and χ(M) is the Euler characteristic of M.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

The Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem illustrates a connection betweentopological and geometric properties.

Theorem: If M is a compact, two-dimensional, Riemannianmanifold with boundary ∂M, then∫

MK dA +

∫∂M

kg ds = 2πχ(M),

Where K is the Gauss curvature on M, kg is the geodesiccurvature on ∂M, and χ(M) is the Euler characteristic of M.

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

Conclusion

Extensions and applications:

I Spacetime models

I Medical imaging

I Industrial engineering

I Mathematical bridge-builder

Modern Geometry

Richard G. Ligo

Introduction

Euclid’sfoundations

Parallel Notions

Beautiful curves

Conclusion

References

I Mlodinow, Leonard (2001). Euclid’s Window. NewYork, NY: The Free Press.

I Oprea, John (2007). Differential Geometry and itsApplications. Washington, DC: The MathematicalAssociation of America.

I Stewart, Ian (2001). Flatterland. Cambridge, MA:Perseus Publishing.

I Wallace, Edward and West, Stephen (2004). Roads toGeometry. Upper Saddle River, NJ: Pearson Education.