Manifolds near the boundary of existence

Abstract

Our primary purpose is to study relationships between bounds on sectional curvature K M , diameter d M , and volume V M , together with their effects on the topology of closed Riemannian ^-manifolds M. For purposes of illustration, we normalize for the moment all manifolds so that d M = π .Consequently, min K M < 1 by the Bonnet-Meyers theorem, and from the Rauch comparison theorem, V M < V(n, minK M , π), where V (n,k,D) denotes the volume of a D-ball, B^(p, D) in the «-dimensional simply connected manifold of constant curvature k.We may thus represent any closed Riemannian /7-manifold as a point in the (minA: M , F^)-plane (Figure 1, next page).Any manifold to the right of a vertical line has a priori bounded Bettinumbers as Gromov proved in [6].Moreover, in regions bounded to the left by a vertical line and below by a horizontal line above the axis, only finitely many homotopy types occur [10].In fact, at least when n Φ 3,4, such regions contain at most finitely many diffeomorphism types [15].Understanding convergence and limit spaces with respect to the Gromov-Hausdorff distance is an essential tool in the proof of the latter result.On the basis of these results, it is natural to examine the topological properties of manifolds close to the boundary of existence.As already explained, the existence region is bounded above by the curve V(n, , π), to the right by the line min K M = 1, and below by the min A^M-axis.Only two points, (1, V(n, 1, π)) and (1/4, V(n, 1/4, π)), on this boundary are actually represented by manifolds, namely the sphere and the real projective space of constant curvature.Moreover, any manifold located strictly between the vertical lines through these two extremal points is homeomorphic to S n [14].

Locations

• Journal of Differential Geometry - View - PDF