The cohomology ring of a smooth manifold

GROUPOID ASSOCIATED TO A SMOOTH MANIFOLD

‎In this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎We show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a Lie groupoid‎. ‎Using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the Lie algebra of all vector fields on the smooth...

groupoid associated to a smooth manifold

‎in this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎we show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a lie groupoid‎. ‎using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the lie algebra of all vector fields on the smooth...

The Real Cohomology Ring of a Sphere Bundle over a Differentiable Manifold

The Lie Algebra of a Smooth Manifold

It is well known that certain topological spaces are determined by rings of continuous real functions defined over them [l; 2; 3],1 and for differentiable manifolds the functions may be differen tiable [4; 7]. In this note we prove that the Lie algebra of all tangent vector fields with compact supports on an infinitely differentiable manifold determines the manifold, and that two such manifolds...

Computing the Poisson Cohomology of a B-poisson Manifold

Because Poisson cohomology is quite challenging to compute, there are only very select cases where the answer is known. In the case of a symplectic manifold where the Poisson bi-vector is non-degenerate, the Poisson cohomology is isomorphic to the de Rham cohomology. The non-degeneracy of π allows us to define an isomorphism T ∗M → TM that provides this isomorphism in cohomology: H(M) ' H π(M)....