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Cercle. --- Surfaces of constant curvature. --- Surfaces à courbure constante.

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Riemannian manifolds. --- Topological degree. --- Hypersurfaces. --- Curvature. --- Riemann, Variétés de. --- Degré topologique. --- Courbure.

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"This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various "nonpositive curvature" and "local-to- global" properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises (1-skeletons, collinearity graphs, covering graphs, domains, etc.) in several seemingly-unrelated fields of mathematics: Metric graph theory; Geometric group theory; Incidence geometries and buildings; Theoretical computer science and combinatorial optimization. We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexed specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With l1-embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT([empty set]) property). Their cells have a specific structure: they are basis polyhedra of even -matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT([empty set]) metric on the orthoscheme complexes of modular lattices; we answer Chastand's question about prime graphs for pre-median graphs. We also explore negative curvature for weakly modular graphs"--

Graph theory. --- Curvature. --- Distance geometry. --- Combinatorial optimization. --- Graphes, Théorie des --- Courbure --- Topologie de l'espace métrique --- Optimisation combinatoire

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Spaces of constant curvature. --- Espaces à courbure constante --- 514.76 --- Geometry of differentiable manifolds and of their submanifolds --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Espaces à courbure constante --- Spaces of constant curvature --- Constant curvature, Spaces of --- Curvature --- Geometry, Differential --- Géometrie différentielle

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"This original text for courses in differential geometry is geared toward advanced undergraduate and graduate majors in math and physics. Based on an advanced class taught by a world-renowned mathematician for more than fifty years, the treatment introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Starting with an introduction to the various curvatures associated to a hypersurface embedded in Euclidean space, the text advances to a brief review of the differential and integral calculus on manifolds. A discussion of the fundamental notions of linear connections and their curvatures follows, along with considerations of Levi-Civita's theorem, bi-invariant metrics on a Lie group, Cartan calculations, Gauss's lemma, and variational formulas. Additional topics include the Hopf-Rinow, Myer's, and Frobenius theorems; special and general relativity; connections on principal and associated bundles; the star operator; superconnections; semi-Riemannian submersions; and Petrov types. Prerequisites include linear algebra and advanced calculus, preferably in the language of differential forms." [Publisher]

Curvature. --- Semi-Riemannian geometry. --- Geometry, Differential. --- Differential calculus. --- Algebras, Linear. --- Relativity (Physics). --- Courbure. --- Géométrie différentielle. --- Calcul différentiel. --- Algèbre linéaire. --- Relativity (Physics) --- Relativité (physique) --- Géométrie de Riemann. --- Géométrie de Riemann. --- Géométrie différentielle.