Download PDF by Dahl M.: A brief introduction to Finsler geometry

By Dahl M.

Show description

Read or Download A brief introduction to Finsler geometry PDF

Best geometry and topology books

Download e-book for kindle: Connections: the geometric bridge between art and science by Jay Kappraff

The 1st version of Connections used to be selected by way of the nationwide organization of Publishers (USA) because the most sensible publication in "Mathematics, Chemistry, and Astronomy - specialist and Reference" in 1991. it's been a entire reference in layout technology, bringing jointly in one quantity fabric from the parts of percentage in structure and layout, tilings and styles, polyhedra, and symmetry.

Additional resources for A brief introduction to Finsler geometry

Example text

Proof. Suppose E is stationary for a curve c. 6 implies that cˆ is a integral curve of G. 4. 8. 17. If γ : I → T M \ {0} is an integral curve of G/F , then π ◦ γ is a stationary curve for E. Conversely, if c is a stationary curve for E, then λ = F ◦ cˆ is constant and c ◦ M1/λ (see below) is an integral curve of G/F . If s > 0, we denote by Ms the mapping Ms : t → st, t ∈ R. 35 Proof. Let c : I → T M \ {0} be an integral curve of G/F . If c = (x, y), then dxi dt dy i dt = yi , λ = −2 Gi ◦ c , λ where λ = F ◦γ > 0 is constant.

14. In a Finsler space (M, F ), the Hilbert 1-form η and the geodesic spray G satisfy 1 dη(G, ·) = d( F 2 ), 2 so X 1 F 2 = G. What is more, XF = G/F . 2 Proof. Using dη = − ∂gij i r y dx ∧ dxj + gij dxi ∧ dy j , ∂xr we obtain dη(G, ·) = = ∂gij ∂gis 1 ∂F 2 i i j i s − dy y y + 2g G dx + is ∂xs ∂xj 2 ∂y i 1 ∂F 2 i 1 ∂F 2 i dx + dy . 2 ∂xi 2 ∂y i The second claim follows since ιXF ω = dF = ιG/F ω. 14 state that dη is preserved under the flow of G. 15. F is a constant on integral curves of G and G/F .

Equation (47) follows from equation (48). The next proposition shows how stationary curves of E are transformed under the Legendre transformation. 19. 18, and π is the canonical projection π : T ∗ M → M . 1. If γ : I → T ∗ M \ {0} is an integral curve of Xh , then π ◦ γ is a stationary curve for E. 36 2. If c : I → M is a stationary curve for E, then λ = F ◦ cˆ is constant and L −1 ◦ c ◦ M1/λ is an integral curve of Xh . Proof. In the first claim, L ◦ γ is an integral curve of X F . 17. The other proof is similar.

Download PDF sample

A brief introduction to Finsler geometry by Dahl M.


by Kevin
4.5

Rated 4.44 of 5 – based on 17 votes