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# On the Parallelism

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```									 Workshop on Finsler Geometry and its Applications Debrecen 2009

On the parallel displacement and parallel
vector fields in Finsler Geometry

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 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. 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

Horizontal parts          ||
０
Vertical parts
§3. HTM

Horizontal parts   ||
０
§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,

∇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)

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

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