Search results
Results from the WOW.Com Content Network
Let be a smooth manifold; a (smooth) distribution assigns to any point a vector subspace in a smooth way. More precisely, consists of a collection {} of vector subspaces with the following property: Around any there exist a neighbourhood and a collection of vector fields, …, such that, for any point , span {(), …, ()} =.
This is true for symplectic manifolds. If the manifold M has a metric of positive scalar curvature and b 2 + (M) ≥ 2 then all Seiberg–Witten invariants of M vanish. If the manifold M is the connected sum of two manifolds both of which have b 2 + ≥ 1 then all Seiberg–Witten invariants of M vanish.
Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k (M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex.
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold.It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology.
Chern, Shiing-Shen (1951), Topics in Differential Geometry, Institute for Advanced Study, mimeographed lecture notes. Chern, Shiing-Shen (1995), Complex Manifolds Without Potential Theory, Springer-Verlag, ISBN 0-387-90422-0, ISBN 3-540-90422-0. (The appendix of this book, "Geometry of Characteristic Classes," is a very neat and profound ...
Topology of 4-manifolds. Princeton Mathematical Series. Vol. 39. Princeton, NJ: Princeton University Press. ISBN 0-691-08577-3. (This does the theorem for topological 4-manifolds.) Milnor, John, Lectures on the h-cobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, Princeton, NJ, 1965. v+116 pp.
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...
Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric. [13] This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact.