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 exotic sphere if it not diffeoemorphic to a standard sphere. | 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 exotic spheres) is given on [[Exotic spheres]]. | + | General information about homotopy spheres (and exotic spheres) is given on [[Exotic spheres]]. One prominent question concerning the geometry of exotic spheres is: |
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− | One prominent question concerning the geometry of exotic spheres is: | + | |
Given an exotic sphere, are there Riemannian metrics which fulfil specific positivity criteria? | Given an exotic sphere, are there Riemannian metrics which fulfil specific positivity criteria? | ||
− | == Homotopy spheres with positive [[Wikipedia: | + | == Homotopy spheres with positive [[Wikipedia:Sectional_curvature|sectional curvature]] == |
− | == Homotopy spheres with positive [[Ricci curvature]] == | + | == Homotopy spheres with positive [[Wikipedia:Ricci_curvature|Ricci curvature]] == |
− | == Homotopy spheres with positive [[scalar curvature]] == | + | == Homotopy spheres with positive [[Wikipedia:Scalar_curvature|scalar curvature]] == |
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1 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 fulfil specific positivity criteria?