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On the parallel displacement and parallel vector fields in Finsler Geometry

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Title: On the parallel displacement and parallel vector fields in Finsler Geometry

1
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

2
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

3
1. Definition of the parallel displacement along
a curve
4
1. Definition of the parallel displacement along
a curve
5
1. Definition of the parallel displacement along
a curve
6
1. Definition of the parallel displacement along
a curve
7
1. Definition of the parallel displacement along
a curve
8
1. Definition of the parallel displacement along
a curve
9
1. Definition of the parallel displacement along
a curve
In this time, we have another curve c-1 and
vector field v-1 .
10
1. Definition of the parallel displacement along
a curve
11
1. Definition of the parallel displacement along
a curve
The vector field v-1 is not parallel along c-1 .
Because
12
1. Definition of the parallel displacement along
a curve
The vector field v-1 is not parallel along c-1 .
Because
13
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,
14
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,
15
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,
16
2. Parallel vector fields on curves c and c-1
So, we take a parallel vector field u on c-1 as
follows
17
2. Parallel vector fields on curves c and c-1
And, we consider the transformation FA ? A on
TpM
18
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.
19
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)
20
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)
21
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)
22
2. Parallel vector fields on curves c and c-1
23
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
24
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
25
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
26
2. Parallel vector fields on curves c and c-1
So we have
27
2. Parallel vector fields on curves c and c-1
So we have
From u(a)v-1(a)B,
28
2. Parallel vector fields on curves c and c-1
So we have
From u(a)v-1(a)B,
So we have
29
3. HTM
We show the geometrical meaning of Definition 1.
30
3. HTM
We show the geometrical meaning of Definition 1.
31
3. HTM
We show the geometrical meaning of Definition 1.
32
3. HTM
33
3. HTM
34
3. HTM
So, we have
Therefore
35
3. HTM
So, we have
Therefore
We can take the derivative operator with respect
to xi
36
3. HTM
37
3. HTM
38
3. HTM
39
3. HTM
40
3. HTM
0
Horizontal parts
Vertical parts
41
3. HTM
0
Horizontal parts
42
4. Paths and Autoparallel curves
Path
43
4. Paths and Autoparallel curves
Path
44
4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one?
45
4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one?
In general, Not !
46
4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one?
In general, Not !
Because
47
4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one?
In general, Not !
Because
48
4. Paths and Autoparallel curves
Path
If c is a path, then is c-1 also the one?
In general, Not !
Because
49
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.
50
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.
51
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.
52
4. Paths and Autoparallel curves
Autoparallel curve
53
4. Paths and Autoparallel curves
Autoparallel curve
Definition 2. The curve c(ci(t)) is called
an autoparallel curve.
54
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.
55
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
56
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.
57
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
58
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
59
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
60
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.
61
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
62
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
63
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
64
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
65
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
66
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
67
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
68
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
0
0
69
5. Inner product
Inversely,
For the parallel vector fields v, u on c,
If and is
constant on c, then we have
70
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 ( )
71
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
72
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.
73
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.
74
6. Geodesics
By using Cartan connection, the equation of a
geodesic c(ci(t)) is
75
6. Geodesics
By using Cartan connection, the equation of a
geodesic c(ci(t)) is
( t is the arc-length.)
76
6. Geodesics
By using Cartan connection, the equation of a
geodesic c(ci(t)) is
( t is the arc-length.)
77
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
78
7. Parallel vector fields
79
7. Parallel vector fields
In the case of Riemannian Geometry
80
7. Parallel vector fields
In the case of Riemannian Geometry
A vector field v(x) on M is parallel, if and
only if,
81
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)
82
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,
83
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
84
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.
85
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
86
7. Parallel vector fields
I want the notion of parallel vector field in
Finsler geometry.
87
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)
88
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.
89
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.
90
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
91
7. Parallel vector fields
So we consider the lift
to HTM.
And calculate the differential with respect to
92
7. Parallel vector fields
So we consider the lift
to HTM.
And calculate the differential with respect to
93
7. Parallel vector fields
So we consider the lift
to HTM.
And calculate the differential with respect to
94
7. Parallel vector fields
So we consider the lift
to HTM.
And calculate the differential with respect to
95
7. Parallel vector fields
So we consider the lift
to HTM.
And calculate the differential with respect to
96
7. Parallel vector fields
So we consider the lift
to HTM.
And calculate the differential with respect to
97
7. Parallel vector fields
So we consider the lift
to HTM.
And calculate the differential with respect to
98
7. Parallel vector fields
So we treat the case satisfying
99
7. Parallel vector fields
So we treat the case satisfying
0
100
7. Parallel vector fields
So we treat the case satisfying
0
0
101
7. Parallel vector fields
So we treat the case satisfying
0
0
Namely,
(7.1)
(7.2)
102
7. Parallel vector fields
So we treat the case satisfying
0
0
Namely,
(7.1)
(7.2)
103
7. Parallel vector fields
First of all
104
7. Parallel vector fields
First of all
105
7. Parallel vector fields
The curve c is called the flow of v.
106
7. Parallel vector fields
The curve c is called the flow of v.
Then the restriction
satisfies
107
7. Parallel vector fields
The curve c is called the flow of v.
Then the restriction
satisfies
108
7. Parallel vector fields
The curve c is called the flow of v.
Then the restriction
satisfies
Because, from (7.2)
109
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.
110
7. Parallel vector fields
Next, we can see the solution c(t) satisfies
111
7. Parallel vector fields
Next, we can see the solution c(t) satisfies
Because, from(7.1)
112
7. Parallel vector fields
Next, we can see the solution c(t) satisfies
Because, from(7.1)
So the curve c is a path.
113
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.
114
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.
115
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.
116
7. Parallel vector fields
Conclusion1
117
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)
118
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),
119
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),
120
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.
121
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
122
7. Parallel vector fields
By the integrability conditions
of (7.2),
123
7. Parallel vector fields
By the integrability conditions
of (7.2),
124
7. Parallel vector fields
By the integrability conditions
of (7.2), the equation
(7.4)
is satisfied.
125
7. Parallel vector fields
By the integrability conditions
of (7.2), the equation
(7.4)
are satisfied.
Because
126
7. Parallel vector fields
By the integrability conditions
of (7.2), the equation
(7.4)
is satisfied.
Because
127
7. Parallel vector fields
Lastly, the solutions of (7.1) and (7.2)
coincide, that is,
128
7. Parallel vector fields
Lastly, the solutions of (7.1) and (7.2)
coincide, that is,
(7.5)
129
7. Parallel vector fields
Lastly, the solutions of (7.1) and (7.2)
coincide, that is,
(7.5)
From
130
7. Parallel vector fields
Lastly, the solutions of (7.1) and (7.2)
coincide, that is,
(7.5)
From
Thus we have
131
8. Comparison to Riemannian cases
132
8. Comparison to Riemannian cases
133
8. Comparison to Riemannian cases
134
8. Comparison to Riemannian cases
135
8. Comparison to Riemannian cases
136
8. Comparison to Riemannian cases
137
8. Comparison to Riemannian cases
138
8. Comparison to Riemannian cases
139
8. Comparison to Riemannian cases
140
8. Comparison to Riemannian cases
141
8. Comparison to Riemannian cases
142
8. Comparison to Riemannian cases
143
References