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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