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

§1. Definition of the parallel displacement along a curve

§1. Definition of the parallel displacement along a curve

§1. Definition of the parallel displacement along a curve

§1. Definition of the parallel displacement along a curve

§1. Definition of the parallel displacement along a curve

§1. Definition of the parallel displacement along a curve

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

§1. Definition of the parallel displacement along a curve

§1. Definition of the parallel displacement along a curve

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

§1. Definition of the parallel displacement along a curve

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

§1. Definition of the parallel displacement along a curve

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

If v-1 is parallel,

§1. Definition of the parallel displacement along a curve

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

If v-1 is parallel,

§1. Definition of the parallel displacement along a curve

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:

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

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

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

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.

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

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

A=(Ai), A’ =(A’i)

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

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

A=(Ai), A’ =(A’i)

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

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

A=(Ai), A’ =(A’i)

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

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

Under this assumption, we can prove that

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

As follows:

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

Under this assumption, we can prove that

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

As follows:

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

Under this assumption, we can prove that

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

As follows:

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

So we have:

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

So we have:

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

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

So we have:

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

So we have:

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

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

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

§3. HTM

§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

§3. HTM

§3. HTM

§3. HTM

||０

Horizontal parts

Vertical parts

§3. HTM

||０

Horizontal parts

§4. Paths and Autoparallel curves

Path

§4. Paths and Autoparallel curves

Path

§4. Paths and Autoparallel curves

Path

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

§4. Paths and Autoparallel curves

Path

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

§4. Paths and Autoparallel curves

Path

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

Because

§4. Paths and Autoparallel curves

Path

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

Because

§4. Paths and Autoparallel curves

Path

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

Because

§4. Paths and Autoparallel curves

Path

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

Because

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

§4. Paths and Autoparallel curves

Path

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

Because

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

§4. Paths and Autoparallel curves

Path

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

Because

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

§4. Paths and Autoparallel curves

Autoparallel curve

§4. Paths and Autoparallel curves

Autoparallel curve

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

§4. Paths and Autoparallel curves

Autoparallel curve

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

In other words,

The canonical lift to HTM is horizontal.

§4. Paths and Autoparallel curves

Autoparallel curve

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

§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,

§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,

§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,

§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

So, if , then is constant on c.

For the parallel vector fields v, u on c,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

|| 0

|| 0

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

V, u : arbitrarily Not Riemannian case ( )

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

V, u : arbitrarily Not Riemannian case ( )

So, we have

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

V, u : arbitrarily Not Riemannian case ( )

So, we have

c is a path.

§5. Inner product

If and is constant on c, then we have

For the parallel vector fields v, u on c,

Inversely,

V, u : arbitrarily Not Riemannian case ( )

So, we have

c is a path.

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

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

( t is the arc-length.)

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

( t is the arc-length.)

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

( t is the arc-length.)

According to the above discussion, we have

§7. Parallel vector fields

§7. Parallel vector fields

In the case of Riemannian Geometry

§7. Parallel vector fields

In the case of Riemannian Geometry

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

§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)

§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,

§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,

Then v(x) has the following properties:

§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,

Then v(x) has the following properties:

(1) v is parallel along any curve c.

§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,

Then v(x) has the following properties:

(1) v is parallel along any curve c.

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

§7. Parallel vector fields

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

§7. Parallel vector fields

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)

§7. Parallel vector fields

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.

§7. Parallel vector fields

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.

§7. Parallel vector fields

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)

And calculate the differential with respect to

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

So we consider the lift to HTM.

§7. Parallel vector fields

And calculate the differential with respect to

So we consider the lift to HTM.

§7. Parallel vector fields

And calculate the differential with respect to

So we consider the lift to HTM.

§7. Parallel vector fields

And calculate the differential with respect to

So we consider the lift to HTM.

§7. Parallel vector fields

And calculate the differential with respect to

So we consider the lift to HTM.

§7. Parallel vector fields

And calculate the differential with respect to

So we consider the lift to HTM.

§7. Parallel vector fields

And calculate the differential with respect to

So we consider the lift to HTM.

§7. Parallel vector fields

And calculate the differential with respect to

So we consider the lift to HTM.

§7. Parallel vector fields

So we treat the case satisfying

§7. Parallel vector fields

So we treat the case satisfying

0

§7. Parallel vector fields

So we treat the case satisfying

0 0

§7. Parallel vector fields

So we treat the case satisfying

0 0

Namely,(7.1)

(7.2)

§7. Parallel vector fields

So we treat the case satisfying

0 0

Namely,(7.1)

(7.2)

§7. Parallel vector fieldsFirst of all

§7. Parallel vector fieldsFirst of all

§7. Parallel vector fields

The curve c is called the flow of v.

§7. Parallel vector fields

The curve c is called the flow of v.

Then the restriction satisfies

§7. Parallel vector fields

The curve c is called the flow of v.

Then the restriction satisfies

§7. Parallel vector fields

The curve c is called the flow of v.

Then the restriction satisfies Because, from (7.2)

§7. Parallel vector fields

The curve c is called the flow of v.

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

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

Because, from(7.1)

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

Because, from(7.1)

So the curve c is a path.

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

Because, from(7.1)

So the curve c is a path.

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

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

Because, from(7.1)

So the curve c is a path.

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

The inner product is constant on c.

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

Because, from(7.1)

So the curve c is a path.

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

The inner product is constant on c.

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

§7. Parallel vector fieldsConclusion1

§7. Parallel vector fields

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)

§7. Parallel vector fields

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),

§7. Parallel vector fields

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),

§7. Parallel vector fields

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), the equation

is satisfied.

(7.3)

§7. Parallel vector fields

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), the equation

is satisfied.

Because

(7.3)

§7. Parallel vector fields

By the integrability conditions of (7.2),

§7. Parallel vector fields

By the integrability conditions of (7.2),

§7. Parallel vector fields

By the integrability conditions of (7.2), the equation

is satisfied.

(7.4)

§7. Parallel vector fields

By the integrability conditions of (7.2), the equation

are satisfied.

Because

(7.4)

§7. Parallel vector fields

By the integrability conditions of (7.2), the equation

is satisfied.

Because

(7.4)

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

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

(7.5)

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

From

(7.5)

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

From

Thus we have

(7.5)

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

§8. Comparison to Riemannian cases

References