Topological obstructions to nonnegative curvature

The purpose of the present note is to prove the following results. Let M be a connected differentiable manifold which satisfies the second axiom of countability. Then (i) M admits a complete Riemannian metric; (ii) If every Riemannian metric on M is complete, M must be compact. In fact, somewhat stronger results will be given as Theorems 1 and 2 below. Let M be a connected differentiable manifold. It is known that if M satisfies the second axiom of countability, then M admits a Riemannian metric. Conversely, it can be shown that the existence of a Riemannian metric on M implies that M satisfies the countability axiom. For any Riemannian metric g on M, we can define a natural metric d on M by setting the distance d(x, y) between two points x and y to be the infinimum of the lengths of all piecewise differentiable curves joining x and y. The Riemannian metric g is complete if the metric space M with d is complete. It is known that this is the case if and only if every bounded subset of M (with respect to d) is relatively compact. We shall say that a Riemannian metric g is bounded if M is bounded with respect to the metric d. We shall prove

The purpose of this paper is to construct complete Riemannian Manifolds with negative Ricci curvature metric. We shall mostly concentrate on threedimensional manifolds. In [BJ], Bourguignon raised the problem that the connected sum of two Riemannian Manifolds with negative Ricci curvature has a metric with negative Ricci curvature. Yau first observed that the connected sum of two hyperbolic manifolds has a negative Ricei curvature metric. In the first part of the paper, we shall give the affirmative answer to the above question. We actually prove something more. We shall prove that the "connected sum along circles" of two Riemannian Manifolds with negative Ricci curvature metric (Sect. 1) has a negative Ricci curvature metric. In the second part, we only consider three-dimensional manifolds. We use the above results and some topological constructions to obtain new manifolds with negative Ricci curvature metric from a given manifold with negative Ricci curvature metric. Give M a complete Riemannian three-dimensional manifold with negative Ricci curvature metric, then there are complete negative Ricci curvature Riemannian metric on M # S 2 x S I ; M ~ S 2 x S I # L ( p , q ) , and M ~ S 2 x S ~ 1; x S 1, where 2; is any oriented Riemann surface. For the motivation of this paper see [YS]. This paper is an extended version of the second part of the author's Ph.D. thesis at Stony Brook. I would like to thank my advisor Professor Blaine Lawson for suggesting this problem to me, and inspiring discussions and valuable advice.

We show that for an orientable non-spin manifold with fundamental group Z2 and universal cover S2×S3, the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The representatives of the components are quotients of the standard metric on S3×S3 or metrics on Brieskorn varieties previously constructed using cohomogeneity one actions. The components are distinguished using the relative η invariant of the spinc Dirac operator computed by means of a Lefschetz fixed point theorem.

Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes forms a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,\pi_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $\epsilon=\epsilon(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-\epsilon, -1, G, n}$ has zero rational Pontrjagin classes.

We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.

This paper addresses J. Cheeger and D. Gromoll's question about\nwhich vector bundles admit a complete metric of nonnegative curvature,\nand it relates their question to the issue of which sphere bundles\nadmit a metric of positive curvature. We show that any vector bundle\nthat admits a metric of nonnegative curvature must admit a connection,\na tensor, and a metric on the base space, which together satisfy a\ncertain differential inequality. On the other hand, a slight\nsharpening of this condition is sufficient for the associated sphere\nbundle to admit a metric of positive curvature. Our results sharpen\nand generalize M. Strake and G. Walschap's conditions under which a\nvector bundle admits a connection metric of nonnegative curvature.

We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups $(H,K,G)$ whether nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachh\"ofer and K. Tapp \cite{ST}, we examine all $G$-invariant fibration metrics on $G/H$ for $G$ a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of nonnegative sectional curvature.

We study geometric properties of the metric projection r: M S of an open manifold M with nonnegative sectional curvature onto a soul S. ir is shown to be C? up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of S also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of M, and stu(dy the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of M. The resolution of the Soul conjecture of Cheeger and Gromoll by Perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected. Orne of the key results in [14] is that the metric projection wF: M -> S which maps a point p in M to the point 7r(p) in S that is closest to p is a Riemannian submersioni. Perelman also observed that this map is at least of class C1, and later it was shown in [9] that 7w is at least C2, and C? at almnost every point. The existence of such a Riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of M inust be special. The purpose of this paper is to illustrate this in several different directions. In fact, we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map -w. The paper is essentially structured as follows: After introducing notation and recalling some basic results in sectioni 1, we establish in section 2 the existence of maps similar to -w for any submanifold of M homologous and isometric to the soul, and use this to prove the existence of convex tubular neighborhoods for them. In particular, this also proves: Theorem 2.5. The unit normal bundle vti(S) admits a metric with nonnegative sectional curvature. As a consequence, open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound (see also [81). Theorem 2.5 also implies that nontrivial plane bundles over a Bieberbach maanifold do not admit nonnegatively curved metrics, thereby providing a shorter proof of the main result in [13]. We are grateful to the referee for pointing out this fact to us. Properties such as these are established via stanidard submersion techniques, but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of M far away from the soul. One way around Received by the editors August 18, 1997. 1991 Mathematics Subject Classification. Primary 53C20.

Conformal deformation of warped products and scalar curvature functions on open manifolds

We study the metric geometry of homogeneous reductive spaces of the unitary group of a finite von Neumann algebra with a non complete Riemannian metric. The main result gives an abstract sufficient condition in order that the geodesics of the Levi-Civita connection are locally minimal. Then, we show how this result applies to several examples.

We study the existence of pluricanonical forms on an algebraic variety of general type with infinite fundamental group. We have a nonvanishing of m th plurigenus of the variety by that of a general fibre of the so-called Shafarevich map of the variety.

Let M be a connected, simply connected, oriented, closed, smooth four-manifold which is spin (or equivalently having even intersection form) and put M×≔M∖{point}. In this paper we prove that if X× is a smooth four-manifold homeomorphic but not necessarily diffeomorphic to M× (more precisely, it carries a smooth structure à la Gompf) then X× can be equipped with a complete Ricci-flat Riemannian metric. As a byproduct of the construction it follows that this metric is self-dual as well consequently X× with this metric is in fact a hyper-Kähler manifold. In particular we find that the largest member of the Gompf–Taubes radial family of large exotic R4’s admits a complete Ricci-flat metric (and in fact it is a hyper-Kähler manifold).These Riemannian solutions are then converted into Ricci-flat Lorentzian ones thereby exhibiting lot of new vacuum solutions which are not accessible by the initial value formulation. A natural physical interpretation of them in the context of the strong cosmic censorship conjecture and topology change is discussed.

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S 3. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.

Chapter 26 - Wall's Finiteness Obstruction