Similar geometry and topology books

The geometry of actual submanifolds in complicated manifolds and the research in their mappings belong to the main complex streams of up to date arithmetic. during this zone converge the options of assorted and complicated mathematical fields equivalent to P. D. E. 's, boundary worth difficulties, precipitated equations, analytic discs in symplectic areas, advanced dynamics.

In den Niederlanden erscheint seit etwa zehn Jahren die erfolgreiche "Zebra-Reihe" mit Broschüren zu Mathematik fuer Projekte und selbstgesteuertes Lernen. Die Themen sind ohne vertiefte Vorkenntnisse zugänglich und ermöglichen eigenes Erforschen und Entdecken von Mathematik. Die Autoren der Bände sind Schulpraktiker, Mathematikdidaktiker und auch Fachwissenschaftler.

Extra resources for 4th Geometry Festival, Budapest

Sample text

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. 37 38 References [AIM94] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto, The theory of sprays and finsler spaces with applications in physics and biology, Fundamental Theories of Physics, Kluwer Academic Publishers, 1994. [Ana96] M. Anastasiei, Finsler Connections in Generalized Lagrange Spaces, Balkan Journal of Geometry and Its Applications 1 (1996), no.

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 . Proof. If c is in integral curve of G, and L is the symplectic mapping induced by F , then L ◦ c is an integral curve of X 1 F 2 ◦L −1 .

Suppose M, N are manifolds, Ψ : M → N is a diffeomorphism. Then the pullback of Ψ for vector fields is the mapping Ψ∗ : X (N ) → X (M ) , Y → (DΨ−1 ) ◦ Y ◦ Ψ. 7. Suppose (M, ω), (N, η) are symplectic manifolds, Φ : M → N is a symplectic mapping such that Φ ∗ η = ω, and h : N → R is a smooth function. Then Φ∗ (Xh ) = Xh◦Φ . What is more, if c : I → N is an integral curve of X h ∈ X (N ), then Φ−1 ◦ c is an integral curve of Xh◦Φ . Proof. The contraction operator satisfies ιΦ∗ X (Φ∗ η) = Φ∗ (ιX η) for all η ∈ Ωk (N ), X ∈ X (N ).