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On the parallel displacement and parallel vector

fields in Finsler Geometry

Workshop on Finsler Geometry and its Applications

Debrecen 2009

- Department of Information and Media Studies
- University of Nagasaki
- Tetsuya NAGANO

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 FA ? A on

TpM

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

And, we consider the transformation FA ? A on

TpM

F is linear because of the linearity of the

equation

In the Riemannian case, F is identity. In

general, in the Finsler case, F is not identity.

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

Then, we have the (1,1)-Finsler tensor field Fi j

with the parameter t

A(Ai), A (Ai)

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

Then, we have the (1,1)-Finsler tensor field Fi j

with the parameter t

A(Ai), A (Ai)

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

Then, we have the (1,1)-Finsler tensor field Fi j

with the parameter t

A(Ai), A (Ai)

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. HTM

We show the geometrical meaning of Definition 1.

3. HTM

We show the geometrical meaning of Definition 1.

3. HTM

We show the geometrical meaning of Definition 1.

3. HTM

3. HTM

3. HTM

So, we have

Therefore

3. HTM

So, 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.

Vertical parts vanish

In other words,

The canonical lift to HTM

is horizontal.

Because

Horizontal parts

Vertical parts

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.

For the parallel vector fields v, u on c,

If c is a path, then we have

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.

For the parallel vector fields v, u on c,

If c is a path, then we have

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.

For the parallel vector fields v, u on c,

If c is a path, then we have

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.

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

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

0

0

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

V, u arbitrarily

Not Riemannian case ( )

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

V, u arbitrarily

Not Riemannian case ( )

So, we have

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

V, u arbitrarily

Not Riemannian case ( )

So, we have

c is a path.

5. Inner product

Inversely,

For the parallel vector fields v, u on c,

If and is

constant on c, then we have

V, u arbitrarily

Not Riemannian case ( )

So, we have

c is a path.

6. Geodesics

By using Cartan connection, the equation of a

geodesic c(ci(t)) is

6. Geodesics

By using Cartan connection, the equation of a

geodesic c(ci(t)) is

( t is the arc-length.)

6. Geodesics

By using Cartan connection, the equation of a

geodesic c(ci(t)) is

( t is the arc-length.)

6. Geodesics

By 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,

?v0. (? 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,

?v0. (? 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,

?v0. (? 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,

?v0. (? 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,

?v0. (? 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,

?T0. (? 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,

?T0. (? 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,

?T0. (? 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,

?T0. (? 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

7. Parallel vector fields

So we consider the lift

to HTM.

And calculate the differential with respect to

7. Parallel vector fields

So we consider the lift

to HTM.

And calculate the differential with respect to

7. Parallel vector fields

So we consider the lift

to HTM.

And calculate the differential with respect to

7. Parallel vector fields

So we consider the lift

to HTM.

And calculate the differential with respect to

7. Parallel vector fields

So we consider the lift

to HTM.

And calculate the differential with respect to

7. Parallel vector fields

So we consider the lift

to HTM.

And calculate the differential with respect to

7. Parallel vector fields

So we consider the lift

to HTM.

And calculate the differential with respect to

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 fields

First of all

7. Parallel vector fields

First 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 fields

Next, we can see the solution c(t) satisfies

7. Parallel vector fields

Next, we can see the solution c(t) satisfies

Because, from(7.1)

7. Parallel vector fields

Next, we can see the solution c(t) satisfies

Because, from(7.1)

So the curve c is a path.

7. Parallel vector fields

Next, 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 fields

Next, 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 fields

Next, 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 fields

Conclusion1

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

(7.3)

is satisfied.

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

(7.3)

is satisfied.

Because

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

(7.4)

is satisfied.

7. Parallel vector fields

By the integrability conditions

of (7.2), the equation

(7.4)

are satisfied.

Because

7. Parallel vector fields

By the integrability conditions

of (7.2), the equation

(7.4)

is satisfied.

Because

7. Parallel vector fields

Lastly, the solutions of (7.1) and (7.2)

coincide, that is,

7. Parallel vector fields

Lastly, the solutions of (7.1) and (7.2)

coincide, that is,

(7.5)

7. Parallel vector fields

Lastly, the solutions of (7.1) and (7.2)

coincide, that is,

(7.5)

From

7. Parallel vector fields

Lastly, the solutions of (7.1) and (7.2)

coincide, that is,

(7.5)

From

Thus we have

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