Curvature properties of exotic spheres
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A homotopy sphere of dimension $n$ is an oriented closed smooth | A homotopy sphere of dimension $n$ is an oriented closed smooth | ||
− | manifold which is homotopy equivalent to the standard sphere $S^n$. A homotopy sphere is called an | + | manifold which is homotopy equivalent to the standard sphere $S^n$. A homotopy sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere. |
− | General information about homotopy spheres (and | + | General information about homotopy spheres (and exotic spheres) is given on [[Theory|Exotic spheres]]. |
+ | |||
+ | One prominent question concerning the geometry of exotic spheres is: | ||
+ | Given an exotic sphere, are there Riemannian metrics which have | ||
+ | positive sectional curvature, positive Ricci Curvature, or positive scalar curvature? | ||
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== Introduction ==
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A homotopy sphere of dimension is an oriented closed smooth manifold which is homotopy equivalent to the standard sphere . A homotopy sphere is called an exotic sphere if it not diffeoemorphic to a standard sphere. General information about homotopy spheres (and exotic spheres) is given on Exotic spheres.
One prominent question concerning the geometry of exotic spheres is: Given an exotic sphere, are there Riemannian metrics which have positive sectional curvature, positive Ricci Curvature, or positive scalar curvature?