Metric Of Nonnegative Sectional Curvature Research Articles

Diffeomorphism of total spaces and equivalence of bundles

Let E 1 and E 2 be the total spaces of smooth, oriented vector bundles of rank k over the n-sphere. We show that if E 1 and E 2 are diffeomorphic, with orientation preserved, then the bundles are smoothly equivalent up to orientation of the base whenever k>[( n+1)/2]+1. With an additional hypothesis, the same conclusion holds when the base is an arbitrary closed, oriented n-manifold. Furthermore, if the base manifold is a homotopy n-sphere and if one of the bundles has a nowhere-zero cross-section, then the oriented bundles are smoothly equivalent up to orientation of the base in the case where k=[( n+1)/2]+1 as well. The latter statement is false if k<[( n+1)/2]+1, as several counterexamples illustrate. We show that each of these examples is an open manifold E admitting a complete metric of nonnegative sectional curvature for which the zero section of the nontrivial vector bundle, a standard sphere, is not the image of a soul in the sense of Cheeger and Gromoll under any diffeomorphism of E.

Einstein manifolds of non-negative sectional curvature and entropy

We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow in terms of the curvature tensor.

A priori bounds for co-dimension one isometric embeddings

Let X : ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], g ) → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] be a C 4 isometric embedding of a C4 metric g of nonnegative sectional curvature on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] into the Euclidean space [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]. We prove a priori bounds for the trace of the second fundamental form H , in terms of the scalar curvature R of g , and the diameter d of the space ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /], g ). These estimates give a bound on the extrinsic geometry in terms of intrinsic quantities. They generalize estimates originally obtained by Weyl for the case n = 2 and positive curvature, and then by P. Guan and the first author for nonnegative curvature and n = 2. Using C 2,α interior estimates of Evans and Krylov for concave fully nonlinear elliptic partial differential equations, these bounds allow us to obtain the following convergence theorem: For any ∈ &gt; 0, the set of metrics of nonnegative sectional curvature and scalar curvature bounded below by ∈ which are isometrically embedable in Euclidean space [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] is closed in the Hölder space C 4,α , 0 &lt; α &lt; 1. These results are obtained in an effort to understand the following higher dimensional version of the Weyl embedding problem which we propose: Suppose that g is a smooth metric of nonnegative sectional curvature and positive scalar curvature on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] which admits locally convex isometric embedding into [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /]. Does ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="09i" /], g ) then admit a smooth global isometric embedding X : ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10i" /], g ) → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="11i" /]?

On complete metrics of nonnegative curvature on 2-plane bundles

This paper is an attempt to understand the 2-plane bundle case for the converse of the soul theorem due to J. Cheeger and D Gromoll. It is shown that there is a class of 2-plane bundles over certain 52-bundles that carry complete metrics of nonnegative sectional curvature. In particular, every 2plane bundle and every S1 -bundle over the connected sum CPnφCPn of CPn with a negative CPn carries a 2-parameter family of complete metrics with nonnegative sectional curvature. A complete noncompact Riemannian manifold with nonnegative curvature (K > 0) is diffeomorphic to the normal bundle of a closed totally geodesic submanifold of the noncompact manifold according to a fundamental theorem due to J. Cheeger and D. Gromoll [4]. They also proposed the following problem: Which vector bundles over closed manifolds with K > 0 carry complete metrics with K > 0? This paper is an attempt to understand the 2-plane bundle case. For a class of 2-plane bundles and S^-bundles, we show that there exist families of complete metrics with K > 0. Our main result is the following Theorem. Every 2-plane bundle and every S1-bundle over the connected sum CP n #CP of CP n with a negative CPn carries a 2-parameter family of complete metrics with K > 0. More generally, let (£?, h) be a Hodge manifold with positive Kahlerian curvature and let M be an associated S2bundle over B. Then, there is a two dimensional subspace H2(M), generated by two integral cohomology classes, in the cohomology group H*(M) such that every principal S1-bundle and every 2-plane bundle over M supported by an Euler class in H 2{M) carries a 2-parameter family of complete metrics with K>0.

Vector bundles with no soul

We exhibit vector bundles over compact manifolds of nonnegative sectional curvature whose total spaces do not admit a complete metric of nonnegative sectional curvature.

Connection metrics of nonnegative curvature on vector bundles

Abstract It is shown that a vector bundleE over a compact positively curved manifold admits complete Riemannian metrics of nonnegative sectional curvature, provided there exists a Riemannian connection onE with almost parallel curvature tensor.

Metric of nonnegative sectional curvature on the tangent bundle to the two-dimensional sphere

Metric of nonnegative sectional curvature on the tangent bundle to the two-dimensional sphere

Curvature, diameter and betti numbers

We give an upper bound for the Betti numbers of a compact Riemannian manifold in terms of its diameter and the lower bound of the sectional curvatures. This estimate in particular shows that most manifolds admit no metrics of non-negative sectional curvature.