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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
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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
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§1. Definition of the parallel displacement along a curve
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§1. Definition of the parallel displacement along a curve
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§1. Definition of the parallel displacement along a curve
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§1. Definition of the parallel displacement along a curve
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§1. Definition of the parallel displacement along a curve
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§1. Definition of the parallel displacement along a curve
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§1. Definition of the parallel displacement along a curve
In this time, we have another curve c-1 and vector field v-1 .
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§1. Definition of the parallel displacement along a curve
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§1. Definition of the parallel displacement along a curve
The vector field v-1 is not parallel along c-1 . Because
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§1. Definition of the parallel displacement along a curve
The vector field v-1 is not parallel along c-1 . Because
![Page 13: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/13.jpg)
§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,
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§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,
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§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,
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§2. Parallel vector fields on curves c and c-1
So, we take a parallel vector field u on c-1 as follows:
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§2. Parallel vector fields on curves c and c-1
And, we consider the transformation Φ :A → A’ on TpM
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§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.
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§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)
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§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)
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§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)
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§2. Parallel vector fields on curves c and c-1
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§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:
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§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:
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§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:
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§2. Parallel vector fields on curves c and c-1
So we have:
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§2. Parallel vector fields on curves c and c-1
So we have:
From u(a)=v-1(a)=B,
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§2. Parallel vector fields on curves c and c-1
So we have:
From u(a)=v-1(a)=B,
So we have:
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§3. HTMWe show the geometrical meaning of Definition 1.
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§3. HTMWe show the geometrical meaning of Definition 1.
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§3. HTMWe show the geometrical meaning of Definition 1.
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§3. HTM
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§3. HTM
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§3. HTMSo, we have
Therefore
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§3. HTMSo, we have
Therefore
We can take the derivative operator with respect to xi
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§3. HTM
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§3. HTM
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§3. HTM
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§3. HTM
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§3. HTM
||0
Horizontal parts
Vertical parts
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§3. HTM
||0
Horizontal parts
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§4. Paths and Autoparallel curves
Path
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§4. Paths and Autoparallel curves
Path
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§4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one?
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§4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one? In general, Not !
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§4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one? In general, Not !
Because
![Page 47: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/47.jpg)
§4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one? In general, Not !
Because
![Page 48: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/48.jpg)
§4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one? In general, Not !
Because
![Page 49: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/49.jpg)
§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.
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§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.
![Page 51: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/51.jpg)
§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.
![Page 52: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/52.jpg)
§4. Paths and Autoparallel curves
Autoparallel curve
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§4. Paths and Autoparallel curves
Autoparallel curve
Definition 2. The curve c=(ci(t)) is called an autoparallel curve.
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§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.
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§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
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§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.
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§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,
![Page 58: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/58.jpg)
§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,
![Page 59: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/59.jpg)
§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,
![Page 60: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/60.jpg)
§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,
![Page 61: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/61.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 62: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/62.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 63: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/63.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 64: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/64.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 65: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/65.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 66: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/66.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 67: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/67.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 68: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/68.jpg)
§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
|| 0
|| 0
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§5. Inner product
If and is constant on c, then we have
For the parallel vector fields v, u on c,
Inversely,
![Page 70: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/70.jpg)
§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 ( )
![Page 71: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/71.jpg)
§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
![Page 72: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/72.jpg)
§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.
![Page 73: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/73.jpg)
§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.
![Page 74: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/74.jpg)
§6. GeodesicsBy using Cartan connection, the equation of a geodesic c=(ci(t)) is
![Page 75: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/75.jpg)
§6. GeodesicsBy using Cartan connection, the equation of a geodesic c=(ci(t)) is
( t is the arc-length.)
![Page 76: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/76.jpg)
§6. GeodesicsBy using Cartan connection, the equation of a geodesic c=(ci(t)) is
( t is the arc-length.)
![Page 77: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/77.jpg)
§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
![Page 78: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/78.jpg)
§7. Parallel vector fields
![Page 79: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/79.jpg)
§7. Parallel vector fields
In the case of Riemannian Geometry
![Page 80: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/80.jpg)
§7. Parallel vector fields
In the case of Riemannian Geometry
A vector field v(x) on M is parallel, if and only if,
![Page 81: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/81.jpg)
§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)
![Page 82: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/82.jpg)
§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,
![Page 83: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/83.jpg)
§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:
![Page 84: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/84.jpg)
§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.
![Page 85: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/85.jpg)
§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
![Page 86: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/86.jpg)
§7. Parallel vector fields
I want the notion of parallel vector field in Finsler geometry.
![Page 87: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/87.jpg)
§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)
![Page 88: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/88.jpg)
§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.
![Page 89: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/89.jpg)
§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.
![Page 90: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/90.jpg)
§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.
![Page 91: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/91.jpg)
§7. Parallel vector fields
And calculate the differential with respect to
So we consider the lift to HTM.
![Page 92: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/92.jpg)
§7. Parallel vector fields
And calculate the differential with respect to
So we consider the lift to HTM.
![Page 93: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/93.jpg)
§7. Parallel vector fields
And calculate the differential with respect to
So we consider the lift to HTM.
![Page 94: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/94.jpg)
§7. Parallel vector fields
And calculate the differential with respect to
So we consider the lift to HTM.
![Page 95: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/95.jpg)
§7. Parallel vector fields
And calculate the differential with respect to
So we consider the lift to HTM.
![Page 96: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/96.jpg)
§7. Parallel vector fields
And calculate the differential with respect to
So we consider the lift to HTM.
![Page 97: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/97.jpg)
§7. Parallel vector fields
And calculate the differential with respect to
So we consider the lift to HTM.
![Page 98: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/98.jpg)
§7. Parallel vector fields
So we treat the case satisfying
![Page 99: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/99.jpg)
§7. Parallel vector fields
So we treat the case satisfying
0
![Page 100: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/100.jpg)
§7. Parallel vector fields
So we treat the case satisfying
0 0
![Page 101: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/101.jpg)
§7. Parallel vector fields
So we treat the case satisfying
0 0
Namely,(7.1)
(7.2)
![Page 102: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/102.jpg)
§7. Parallel vector fields
So we treat the case satisfying
0 0
Namely,(7.1)
(7.2)
![Page 103: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/103.jpg)
§7. Parallel vector fieldsFirst of all
![Page 104: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/104.jpg)
§7. Parallel vector fieldsFirst of all
![Page 105: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/105.jpg)
§7. Parallel vector fields
The curve c is called the flow of v.
![Page 106: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/106.jpg)
§7. Parallel vector fields
The curve c is called the flow of v.
Then the restriction satisfies
![Page 107: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/107.jpg)
§7. Parallel vector fields
The curve c is called the flow of v.
Then the restriction satisfies
![Page 108: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/108.jpg)
§7. Parallel vector fields
The curve c is called the flow of v.
Then the restriction satisfies Because, from (7.2)
![Page 109: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/109.jpg)
§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.
![Page 110: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/110.jpg)
§7. Parallel vector fieldsNext, we can see the solution c(t) satisfies
![Page 111: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/111.jpg)
§7. Parallel vector fieldsNext, we can see the solution c(t) satisfies
Because, from(7.1)
![Page 112: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/112.jpg)
§7. Parallel vector fieldsNext, we can see the solution c(t) satisfies
Because, from(7.1)
So the curve c is a path.
![Page 113: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/113.jpg)
§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.
![Page 114: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/114.jpg)
§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.
![Page 115: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/115.jpg)
§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.
![Page 116: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/116.jpg)
§7. Parallel vector fieldsConclusion1
![Page 117: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/117.jpg)
§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)
![Page 118: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/118.jpg)
§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),
![Page 119: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/119.jpg)
§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),
![Page 120: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/120.jpg)
§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)
![Page 121: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/121.jpg)
§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)
![Page 122: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/122.jpg)
§7. Parallel vector fields
By the integrability conditions of (7.2),
![Page 123: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/123.jpg)
§7. Parallel vector fields
By the integrability conditions of (7.2),
![Page 124: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/124.jpg)
§7. Parallel vector fields
By the integrability conditions of (7.2), the equation
is satisfied.
(7.4)
![Page 125: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/125.jpg)
§7. Parallel vector fields
By the integrability conditions of (7.2), the equation
are satisfied.
Because
(7.4)
![Page 126: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/126.jpg)
§7. Parallel vector fields
By the integrability conditions of (7.2), the equation
is satisfied.
Because
(7.4)
![Page 127: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/127.jpg)
§7. Parallel vector fieldsLastly, the solutions of (7.1) and (7.2) coincide, that is,
![Page 128: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/128.jpg)
§7. Parallel vector fieldsLastly, the solutions of (7.1) and (7.2) coincide, that is,
(7.5)
![Page 129: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/129.jpg)
§7. Parallel vector fieldsLastly, the solutions of (7.1) and (7.2) coincide, that is,
From
(7.5)
![Page 130: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/130.jpg)
§7. Parallel vector fieldsLastly, the solutions of (7.1) and (7.2) coincide, that is,
From
Thus we have
(7.5)
![Page 131: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/131.jpg)
§8. Comparison to Riemannian cases
![Page 132: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/132.jpg)
§8. Comparison to Riemannian cases
![Page 133: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/133.jpg)
§8. Comparison to Riemannian cases
![Page 134: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/134.jpg)
§8. Comparison to Riemannian cases
![Page 135: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/135.jpg)
§8. Comparison to Riemannian cases
![Page 136: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/136.jpg)
§8. Comparison to Riemannian cases
![Page 137: On the parallel displacement and parallel vector fields in Finsler Geometry Department of Information and Media Studies University of Nagasaki Tetsuya](https://reader036.vdocuments.us/reader036/viewer/2022062423/5697bfdc1a28abf838cb11c6/html5/thumbnails/137.jpg)
§8. Comparison to Riemannian cases
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§8. Comparison to Riemannian cases
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§8. Comparison to Riemannian cases
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§8. Comparison to Riemannian cases
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§8. Comparison to Riemannian cases
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§8. Comparison to Riemannian cases
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References