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But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used. The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and it is a manifold, but not a closed manifold.
Its Euler characteristic is 0, by the product property. More generally, any compact parallelizable manifold, including any compact Lie group, has Euler characteristic 0. [13] The Euler characteristic of any closed odd-dimensional manifold is also 0. [14] The case for orientable examples is a corollary of Poincaré duality.
The Poincaré lemma states that if B is an open ball in R n, any closed p-form ω defined on B is exact, for any integer p with 1 ≤ p ≤ n. [1] More generally, the lemma states that on a contractible open subset of a manifold (e.g., ), a closed p-form, p > 0, is exact. [citation needed]
In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem [1] [2] [3] after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain ...
A closed interval [,] is a simple example of a one-dimensional manifold with boundary. Its boundary is the set consisting of the two points a {\displaystyle a} and b {\displaystyle b} . Integrating f {\displaystyle f} over the interval may be generalized to integrating forms on a higher-dimensional manifold.
A closed manifold is a compact manifold without boundary, not necessarily connected. An open manifold is a manifold without boundary (not necessarily connected), with no compact component. For instance, [ 0 , 1 ] {\displaystyle [0,1]} is a compact manifold, S 1 {\displaystyle S^{1}} is a closed manifold, and ( 0 , 1 ) {\displaystyle (0,1)} is ...
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the (n − k) th homology group of M, for all integers k
Every homotopy sphere (a closed n-manifold which is homotopy equivalent to the n-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard n-sphere.