On the parallel displacement and parallel vector fields in Finsler Geometry Department of...

On the parallel displacement and parallel vector fields in Finsler Geometry

Department of Information and Media StudiesUniversity of Nagasaki

Tetsuya NAGANO

Workshop on Finsler Geometry and its Applications Debrecen 2009

Contents

• §1. Definition of the parallel displacement along a curve

• §2. Parallel vector fields on curves c and c-1

• §3. HTM• §4. Paths and Autoparallel curves• §5. Inner Product 　• §6. Geodesics• §7. Parallel vector fields• §8. Comparison to Riemannian cases• References

§1. Definition of the parallel displacement along a curve


In this time, we have another curve c-1 and vector field v-1 .


The vector field v-1 is not parallel along c-1 . Because


The vector field v-1 is not parallel along c-1 . Because

If v-1 is parallel,

§2. Parallel vector fields on curves c and c-1

So, we take a parallel vector field u on c-1 as follows:


And, we consider the transformation Φ ：Ａ　→　Ａ’ on TpM


And, we consider the transformation Φ ：Ａ　→　Ａ’ on TpM

Φ is linear because of the linearity of the equation

In the Riemannian case, Φ is identity.In general, in the Finsler case, Φ is not identity.


Then, we have the (1,1)-Finsler tensor field Φi j with the parameter t

A=(Ai), A’ =(A’i)


Under this assumption, we can prove that

The vector field v-1 is parallel along the curve c-1.

As follows:


So we have:


So we have:

From u(a)=v-1(a)=B,


So we have:

From u(a)=v-1(a)=B,

So we have:

§3. HTMWe show the geometrical meaning of Definition 1.

§3. HTM

§3. HTMSo, we have

Therefore

§3. HTMSo, we have

Therefore

We can take the derivative operator with respect to xi

§3. HTM

§3. HTM

||０

Horizontal parts

Vertical parts

§3. HTM

||０

Horizontal parts

§4. Paths and Autoparallel curves

Path


Path

If c is a path, then is c-1 also the one?


Path

If c is a path, then is c-1 also the one? In general, Not !


Path


Because


Path


Because

However, if satisfies, c-1 is the path.


Autoparallel curve


Autoparallel curve

Definition 2. The curve c=(ci(t)) is called an autoparallel curve.


Autoparallel curve


In other words,

The canonical lift to HTM is horizontal.


Autoparallel curve


In other words,

The canonical lift to HTM is horizontal.

Horizontal parts Vertical parts

Vertical parts vanish

Because

§5. Inner product In here, we call it the “inner product” on the curve c=(ci(t))

where the vector fields v=(vi(t)), u=(ui(t)) are on c.



If c is a path, then we have

For the parallel vector fields v, u on c,



If c is a path, then we have

So, if , then is constant on c.


§5. Inner product

If and is constant on c, then we have


Inversely,

§5. Inner product



Inversely,

|| 0

|| 0

§5. Inner product



Inversely,

§5. Inner product



Inversely,

V, u : arbitrarily Not Riemannian case ( )

§5. Inner product



Inversely,


So, we have

§5. Inner product



Inversely,


So, we have

c is a path.

§6. GeodesicsBy using Cartan connection, the equation of a geodesic c=(ci(t)) is


( t is the arc-length.)


( t is the arc-length.)

According to the above discussion, we have

§7. Parallel vector fields


In the case of Riemannian Geometry



A vector field v(x) on M is parallel, if and only if,




∇ ｖ＝０ . 　　（∇： a connection)





In locally,





In locally,

Then v(x) has the following properties:





In locally,


(1) v is parallel along any curve c.





In locally,


(1) v is parallel along any curve c.

(2) The norm ||v|| is constant on M


I want the notion of parallel vector field in Finsler geometry.



In general, Finsler tensor field T is parallel, if and only if,

∇T ＝０ . 　　（∇： a Finsler connection)





But it is not good to obtain the interesting notion like the Riemannian case.






So we consider the lift to HTM.





And calculate the differential with respect to




So we treat the case satisfying



0



0 0



0 0

Namely,(7.1)

(7.2)

§7. Parallel vector fieldsFirst of all


The curve c is called the flow of v.



Then the restriction satisfies



Then the restriction satisfies Because, from (7.2)



Then the restriction satisfies Because, from (7.2)

So we can call v parallel along the curve c.

§7. Parallel vector fieldsNext, we can see the solution c(t) satisfies


Because, from(7.1)


Because, from(7.1)

So the curve c is a path.


Because, from(7.1)


Thus, v is a parallel vector field along the path c.


Because, from(7.1)



The inner product is constant on c.


Because, from(7.1)



The inner product is constant on c.

The norm ||v|| is constant on c.

§7. Parallel vector fieldsConclusion1


Next, we study the conditions in order for the vector field satisfying (7.1) and (7.2) to exist in locally at every point (x,y)



By the integrability conditions of (7.1),



By the integrability conditions of (7.1), the equation

is satisfied.

(7.3)




is satisfied.

Because

(7.3)


By the integrability conditions of (7.2),



is satisfied.

(7.4)



are satisfied.

Because

(7.4)



is satisfied.

Because

(7.4)

§7. Parallel vector fieldsLastly, the solutions of (7.1) and (7.2) coincide, that is,


(7.5)


From

(7.5)


From

Thus we have

(7.5)

§8. Comparison to Riemannian cases

References

On the parallel displacement and parallel vector fields in Finsler Geometry Department of...

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