Download PDF by Demeter Krupka;Olga Krupkova;Jan Slovak: Differential geometry and its applications: proceedings of

By Demeter Krupka;Olga Krupkova;Jan Slovak

ISBN-10: 9812790608

ISBN-13: 9789812790606

This quantity includes invited lectures and chosen study papers within the fields of classical and sleek differential geometry, international research, and geometric equipment in physics, awarded on the tenth foreign convention on Differential Geometry and its purposes (DGA2007), held in Olomouc, Czech Republic. The ebook covers contemporary advancements and the most recent ends up in the subsequent fields: Riemannian geometry, connections, parabolic geometries, jets, differential invariants, the calculus of adaptations on manifolds, differential equations, Finsler buildings, and geometric tools in physics. it's also a party of the three hundredth anniversary of the delivery of 1 of the best mathematicians, Leonhard Euler, and comprises the Euler lecture "Leonhardt Euler - three hundred years on". The impressive participants contain: A Asada, Manuel de Leon, Jurgen Eichhorn, J-H Eschenburg, and Ivan Kolar.

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Extra info for Differential geometry and its applications: proceedings of the 10th International Conference, DGA 2007, Olomouc, Czech Republic, 27-31 August 2007

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We shall further determine We denote zαi = ∂uα the volume of the body Q ⊂ Tϕ(u) H bounded by the induced on Tϕ(u) H ˜ indicatrix from M ∂Q = Tϕ(u) H ∩ {y ∈ Tϕ(u) Rn+1 |F˜ (y) = 1}. ∂ ∂ ∈ Tu D, then ϕ∗,u (v) = zαi v α ∂uα ∂y i at some fixed point u ∈ D, Q is given by If v = v α ϕ(u) ∈ Tϕ(u) H and hence Q = {v ∈ Tu D | F˜ (ϕ(u), ϕ∗,u (v)) ≤ 1}. In particular, Q is explicitly described as follows: a) In the Randers case (2) we have QR : (y 1 )2 + · · · + (y n+1 )2 ≤ (1 − by n+1 )2 ⇔ y i = zαi v α , i = 1, n + 1 n ⇔ i=1 (zαi v α )2 + (1 − b2 )(zαn+1 v α )2 + 2bzαn+1 v α − 1 ≤ 0.

Znn+1 ) and ˆ = H ˜ νbt H νb ρ n+1 ˜ ab ) (h )α a,b=1,n ν(zα = ν(zαn+1 )αt ρ . (6) After performing an orthogonal transformation, the bounding quadric ∂Q has a canonic analytic equation; hence n n λa (v a )2 + Q: a=1 ∆ ≤0⇔ δ∗ a=1 ( (v a )2 ≤ 1, (v 1 , . . , v n ) ∈ Rn , −∆/(δ∗ λa ))2 ˜ δ∗ = λ1 · · · · · λn = det H ˜ and where λ1 , . . , λn are the eigenvalues of H, ˆ ∆ = det H. Hence V ol(Q) = V ol(Bn ) · (−∆)n V ol(Bn ) · (−∆)n/2 = (n+1)/2 δ∗n · λ1 · · · · · λn δ∗ (7) ˜ αβ , or, denoting η∗ = ρ − ν 2 zαn+1 zβn+1 h V ol(Q) = V ol(Bn ) · (−η∗ )n/2 √ .

For further details, we refer the reader to the original papers Refs. 1–3. Keywords: Maximal surfaces, Lorentzian product spaces, Parabolicity. MS classification: 53C42, 53C50. 1. Introduction The study of maximal surfaces, that is, spacelike surfaces with zero mean curvature, has been of increasing interest in recent years from both physical and mathematical points of view. Here by spacelike we mean that the induced metric from the ambient Lorentzian metric is a Riemannian metric on the surface and the term maximal comes from the fact that these surfaces locally maximize area among all nearby surfaces having the same boundary.

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Differential geometry and its applications: proceedings of the 10th International Conference, DGA 2007, Olomouc, Czech Republic, 27-31 August 2007 by Demeter Krupka;Olga Krupkova;Jan Slovak


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