locally isomorphic sentence in Hindi
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- Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups.
- A tropical curve is defined to be a metric space that is locally isomorphic to a star shaped metric graph.
- 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.
- The only groups for which Mostow rigidity does not hold are all groups locally isomorphic to \ mathrm { PSL } _ 2 ( \ mathbb R ).
- 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.
- Let G be a semisimple Lie group and \ Gamma \ subset G a locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ).
- 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.
- 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.
- 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.
- 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.
- The first condition for the theorem is that the unified group " G contains a subgroup locally isomorphic to the Poincare group . " Therefore the theorem only makes a statement about the unification of the Poincare group with an internal symmetry group.
- Local rigidity holds for lattices in semisimple Lie groups providing the latter have no factor of type A1 ( i . e . locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ) ) or the former is irreducible.
- Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces ( that are locally isomorphic to the Cayley projective plane ), and the second does not occur at all as a holonomy group.
- In this view, the general procedure for solving an equivalence problem is to construct a system of invariants for the " G "-structure which are then sufficient to determine whether a pair of " G "-structures are locally isomorphic or not.
- The Mostow rigidity theorem states that for lattices in simple Lie groups not locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ) ( the group of 2 by 2 matrices with determinant 1 ) any isomorphism of lattices is essentially induced by an isomorphism between the groups themselves.
- Mostow rigidity holds ( in its geometric formulation ) more generally for fundamental groups of all complete, finite volume locally symmetric spaces of dimension at least 3, or in its algebraic formulation for all lattices in simple Lie groups not locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ).
- :" If G is a simple Lie group not locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ) or \ mathrm { SL } _ 2 ( \ mathbb C ) with a fixed Haar measure and v > 0 there are only finitely many lattices in G of covolume less than v ."
- The simplest statement is when \ Gamma is a lattice in a simple Lie group g and the latter is not locally isomorphic to \ mathrm { SL } _ 2 ( \ mathbb R ) or \ mathrm { SL } _ 2 ( \ mathbb C ) and \ Gamma ( this means that its Lie algebra is not that of one of these two groups ).
- If X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a "'( G, X )-structure "'on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a ( G, X )-structures are always manifolds and are called "'( G, X )-manifolds " '.
- For example, the " p "-torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementary abelian group scheme of order " p " 2, but over "'F "'p, it is a finite flat group scheme of order " p " 2 that has either " p " connected components ( if the curve is ordinary ) or one connected component ( if the curve is supersingular ).
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