New PDF release: Analytische Geometrie

By Pickert G.

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The most obvious kind of example is a quotient of a manifold by a locally linear action of a finite group. But not every orbifold, not even every compact smooth orbifold, is a quotient of a manifold by a finite group action. (The simplest counterexample or “bad orbifold” is the “teardrop” X, shown in Figure 3. Here the bottom half of the space is a hemisphere, and the top half is the quotient of a hemisphere by a cyclic group acting by rotations around the pole. ) Figure 3. The teardrop On a smooth orbifold X, we have a notion of Riemannian metric, which on a patch looking like Rn /G, G finite, is simply a Riemannian metric on Rn invariant under the action of G.

As we remarked earlier, the bundle VM has a natural flat connection. If we use this connection to define DV , then Lichnerowicz’s identity (2) will still hold with DV in place of D, since there is no contribution from the curvature of the bundle. Thus κ > 0 implies Ind DV = A(u∗ ([D])) = 0. Thus if A is injective, we can conclude that u∗ ([D]) = 0 in KOn (Bπ). 7. See [80] for details. 9 (Gromov-Lawson). A closed aspherical manifold cannot admit a metric of positive scalar curvature. 9, at least for spin manifolds.

32–33. MR 54 #8741 2. Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR 57 #3310 , Erratum: “A geometric construction of the discrete series for semisimple Lie groups” 3. [Invent. Math. 42 (1977), 1–62; MR 57 #3310 ], Invent. Math. 54 (1979), no. 2, 189–192. MR 81d:22015 4. Paul Baum and Alain Connes, Geometric K-theory for Lie groups and foliations, Enseign. Math. (2) 46 (2000), no. 1-2, 3–42. MR 2001i:19006 5.

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Analytische Geometrie by Pickert G.


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