Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups.
2.
A tropical curve is defined to be a metric space that is locally isomorphic to a star shaped metric graph.
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There is thus a bijective correspondence between complete locally symmetric spaces locally isomorphic to X and of finite Riemannian volume, and torsion-free lattices in G.
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The only groups for which Mostow rigidity does not hold are all groups locally isomorphic to \ mathrm { PSL } _ 2 ( \ mathbb R ).
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By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic.
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Let G be a semisimple Lie group and \ Gamma \ subset G a locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ).
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This Lie group is not determined uniquely; however, any two connected Lie groups with the same Lie algebra are " locally isomorphic ", and in particular, have the same universal cover.
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There are two common ways to define algebraic spaces : they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes.
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A contact analogue of the Darboux theorem holds : all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.
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Scheme-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous ( or differentiable, or complex-analytic, etc . ) functions on Euclidean space.
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