By Stefan Haesen, Leopold Verstraelen
ISBN-10: 9462392390
ISBN-13: 9789462392397
ISBN-10: 9462392404
ISBN-13: 9789462392403
A number of introductory articles is equipped on quite a lot of themes, together with variational difficulties on curves and surfaces with anisotropic curvature. specialists within the fields of Riemannian, Lorentzian and call geometry current cutting-edge reports in their subject matters. The contributions are written on a graduate point and comprise prolonged bibliographies. the 10 chapters are the results of numerous doctoral classes which have been held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.
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The first one arises from the study of Brownian motion defined on Riemannian manifolds, and the second one arises from the theory of electricity. 15 The Brownian motion. The irregular movement of microscopic particles suspended in a liquid, (described by the priest and botanical R.
6 Let K be an infinitesimal symmetry of (M, g, V ). An integral curve of K is a trajectory of (M, g, V ) is and only if it passes through a critical point of GK . Proof Let p ∈ M be a point and let x : [0, b] → M the integral curve of K such that x(0) = p. Assume that x is a trajectory of (M, g, V ), then: 26 R. Giambò and P. Piccione D x(0) ˙ dt = ∇ K p K = −∇V ( p). , p is a critical point of G K . The converse is established by the same argument, using the observation that if x is an integral line of K passing through some critical point of G K , then every point of x is critical for G K .
1 Sym(M, g, V ) is the Lie algebra of the Lie subgroup Sym(M, g, V ) and Sym(M, g, ω) is the Lie algebra of the Lie subgroup Sym(M, g, ω). Proof Left to the reader as an exercise. Thus, the conservative dynamical system (M, g, V ) admits a non trivial symmetry if and only if dim Sym(M, g, V ) > 0; similarly, the magnetic dynamical system (M, g, ω)) admits a non trivial symmetry if and only if dim Sym(M, g, ω) > 0. 6 suggest that we can look for periodic trajectories of (M, g, V ) and (M, g, ω) that are closed integral curves of an infinitesimal symmetry K .
Topics in Modern Differential Geometry by Stefan Haesen, Leopold Verstraelen
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