By Stefan Haesen, Leopold Verstraelen
A number of introductory articles is supplied on quite a lot of issues, together with variational difficulties on curves and surfaces with anisotropic curvature. specialists within the fields of Riemannian, Lorentzian and get in touch with geometry current cutting-edge stories in their subject matters. The contributions are written on a graduate point and comprise prolonged bibliographies. the 10 chapters are the results of quite a few doctoral classes that have been held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.
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2 Symmetries of Exact Magnetic Fields Let (M, g, ω) be an exact magnetic dynamical system and let B be the dual vector field to ω. , if the pull-back φ∗ (ω) coincides with ω. It is easy to show that if φ is a symmetry of (M, g, ω) and z : [0, L] → M is a trajectory of (M, g, ω), then also φ ◦ z is a trajectory of (M, g, ω). Let Sym(M, g, ω) denote the subset of Iso(M, g) consisting of symmetries of (M, g, ω). Clearly, Sym(M, g, ω) is a subgroup of Iso(M, g), and it is closed (relatively to the topology of C 1 -convergence on compact subsets).
As an example, consider M to be the Klein bottle obtained as the quotient of R2 endowed with the Euclidean metric dx 2 + dy 2 by the action of the group generated by the isometries (x, y) → (x + 1, y) and (x, y) → (1 − x, y + 1). The vector field K = ∂∂y on M is Killing; all its integral lines are periodic. It is easily seen that in this case the S1 -action induced by the flow of K has exactly two exceptional orbits, and that the orbit space M/S1 is homeomorphic to the closed interval [0, 21 ].
Noether, Invariante Variationsprobleme, Nachr. Akad. Wiss. -Phys. Kl. II, 1918, 235–257 19. B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity (Academic Press, New York, 1983) 20. P. Piccione, A. Zeghib, Action of discrete groups on stationary Lorentz manifolds. Ergod Theor Dyn Syst. 34(5), 1640–1673 (2014) 21. M. Sánchez, Lorentzian manifolds admitting a Killing vector field. Proc. Second World Congr. Nonlinear Analysts Part 1 (Athens, 1996). Nonlinear Anal. 30(1), 643–654 (1997) 22.
Topics in Modern Differential Geometry by Stefan Haesen, Leopold Verstraelen