The approach uses noncommutative spaces that are close to being ordinary manifolds, or more precisely a class of spaces known as almost-commutative geometries, which are locally a product of a smooth manifold
and a finite noncommutative geometry.
In Section 2, expanding a brief account in [7, Section 5], we give a definition of the fixed-point index following the pattern of Dold's construction [10,11,12] of the index for single-valued maps on ENRs, treating the finite cover p : [~.X] [right arrow] X as the 0-dimensional special case of a fibrewise smooth manifold
p : [~.X] [right arrow] X over a compact ENR X with each fibre a closed manifold of some fixed dimension m.
In 1982, Hamilton  introduced the notion of Ricci flow to find a canonical metric on a smooth manifold
. Then Ricci flow has become a powerful tool for the study of Riemannian manifolds, especially for those manifolds with positive curvature.
We follow the conventional definition of manifolds as local-ringed spaces and, by analogy with smooth manifolds
[14,15] and [Z.sub.2]-graded manifolds [5,8,9], define an N-graded manifold as a local-ringed space which is a sheaf in local N-graded commutative rings on a finite-dimensional real smooth manifold
Z (Definition 47).
A smooth manifold
X with action of a real reductive group G is real spherical if a minimal parabolic subgroup P of G has an open orbit in X.
Let M be a smooth manifold
. The derivatives of a function f : M [right arrow] R along the vector fields X, Y, and Z are defined by
A Jacobi manifold is a smooth manifold
M equipped with a bivector field [pi] and a vector field E such that
Authors examine groups of diffeomorphisms on a smooth manifold
. Their interest stems from V.
Definition 4.2 Let W be a closed subset of a smooth manifold
M which has been decomposed into a finite union of locally closed subsets
Here's his handy explanation of the winning entry: "Chas and Sullivan discovered that the homology of the space of free loops (string topology) of a closed oriented smooth manifold
has a structure of a Batalin-Vilkovisky (BV) algebra.
Proof: Since v points inward on [Delta]M, a smooth manifold
D [subset] M can be found that is diffeomorphic to a two-dimensional disc with v pointing inward on [Delta]D [Nishimura, 1981].