Stability problems in a theorem of F. Schur

Schur's theorem states that an isotropic Riemannian manifold of dimension greater than two has constant curvature. It is natural to guess that compact almost isotropic Riemannian manifolds of dimension greater than two are close to spaces of almost constant curvature. We take the curvature anisotropy as the discrepancy of the sectional curvatures at a point. The main result of this paper is that Riemannian manifolds in Cheeger's class ℜ(n,d,V,A) withL ₁-small integral anisotropy haveL _p-small change of the sectional curvature over the manifold. We also estimate the deviation of the metric tensor from that of constant curvature in theW _p ² -norm, and prove that compact almost isotropic spaces inherit the differential structure of a space form. These stability results are based on the generalization of Schur' theorem to metric spaces.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Conformal deformations of integral pinched 3-manifolds

In this paper we prove that, under an explicit integral pinching assumption between the L²-norm of the Ricci curvature and the L²-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits a conformal metric of positive Ricci curvature. In particular, using a result of Hamilton, this implies that the manifold is diffeomorphic to a quotient of S³. The proof of the main result of the paper is based on ideas developed in an article by M. Gursky and J. Viaclovsky.     

Sur la forme de la boule unité de la norme stable unidimensionnelle

We study the stable norm on the first homology of a Riemannian polyhedron. In the one-dimensional case (metric graphs), the geometry of the unit ball of this norm is completely described by the combinatorial structure of the graph. For a smooth manifold of dimension ≥3 and using polyhedral techniques, we show that a large class of polytopes appears as unit ball of the stable norm associated to some metric conformal to a given one. Received: 18 March     

On tangent cones of Alexandrov spaces with curvature bounded below

The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric. Received: 20 October 1999 / Revised version: 8 May 2000     

Warped products of Hadamard spaces

In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved by warped products with convex warping functions. Received: 9 January 1998/ Revised version: 12 March 1998     

Minimal 2-spheres in a complex projective space

In this paper conformal minimal 2-spheres immersed in a complex projective space are studied by applying Lie theory and moving frames. We give differential equations of Kähler angle and square length of the second fundamental form. By applying these differential equations we give characteristics of conformal minimal 2-spheres of constant Kähler angle and obtain pinching theorems for curvature. We also discuss conformal minimal 2-spheres of constant normal curvature and prove that there does not exist any linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with non-positive constant normal curvature. We also prove that a linearly full minimal 2-sphere immersed in a complex projective space CPn (n>2) with constant normal curvature and constant Kähler angle is of constant curvature.     

Cohomogeneity one Riemannian manifolds of non-positive curvature

We study a G-manifold M which admits a G-invariant Riemannian metric g of non-positive curvature. We describe all such Riemannian G-manifolds (M,g) of non-positive curvature with a semisimple Lie group G which acts on M regularly and classify cohomogeneity one G-manifolds M of a semisimple Lie group G which admit an invariant metric of non-positive curvature. Some results on non-existence of invariant metric of negative curvature on cohomogeneity one G-manifolds of a semisimple Lie group G are given.     

A non-local flow for Riemann surfaces

A non-local flow is defined for compact Riemann surfaces. Assuming the initial metric has positive Gauss curvature and is not conformal to the round sphere, the flow exists on some maximal time interval, and converges along a subsequence to a metric which admits a conformal Killing vector field. By a result of Tashiro (Trans Am Math Soc 117:251–275, 1965), the limiting metric must be conformal to the round sphere. Research supported in part by NSF Grant DMS-0500538.     

Conformally Kähler base metrics for Einstein warped products

A Riemannian metric g with Ricci curvature r is called nontrivial quasi-Einstein, in a sense given by Case, Shu and Wei, if it satisfies (−a/f)∇df+r=λg, for a smooth nonconstant function f and constants λ and a>0. If a is a positive integer, it was noted by Besse that such a metric appears as the base metric for certain warped Einstein metrics. This equation also appears in the study of smooth metric measure spaces. We provide a local classification and an explicit construction of Kähler metrics conformal to nontrivial quasi-Einstein metrics, subject to the following conditions: local Kähler irreducibility, the conformal factor giving rise to a Killing potential, and the quasi-Einstein function f being a function of the Killing potential. Additionally, the classification holds in real dimension at least six. The metric, along with the Killing potential, form an SKR pair, a notion defined by Derdzinski and Maschler. It implies that the manifold is biholomorphic to an open set in the total space of a CP¹ bundle whose base manifold admits a Kähler-Einstein metric. If the manifold is additionally compact, it is a total space of such a bundle or complex projective space. Additionally, a result of Case, Shu and Wei on the Kähler reducibility of nontrivial Kähler quasi-Einstein metrics is reproduced in dimension at least six in a more explicit form.     

Sharp estimates and the Dirac operator

This paper establishes and extends a conjecture posed by M. Gromov which states that every riemannian metric on that strictly dominates the standard metric must have somewhere scalar curvature strictly less than that of . More generally, if is any compact spin manifold of dimension which admits a distance decreasing map of non-zero degree, then either there is a point with normalized scalar curvature , or is isometric to . The distance decreasing hypothesis can be replaced by the weaker assumption is contracting on -forms. In both cases, the results are sharp. An explicit counterexample is given to show that the result is no longer valid if one replaces 2-forms by -forms with . Received: 16 May 1996     

Harmonic maps from closed Riemannian manifolds with positive scalar curvature

It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorem cannot hold in general. This raises the question: “What information can we obtain from the existence of non-constant harmonic map?” This paper gives answer to this problem; the results obtained are optimal.     

Total excess on length surfaces

We study the space of directions on a length space and examine examples having particular spaces of directions. Then we generalize the notion of total excess on length spaces satisfying some suitable conditions, which we call good surfaces. For good surfaces we generalize the Euler characteristic, and prove the generalized Gauss-Bonnet Theorem and other relations between the total excess and the Euler characteristic. Furthermore, we see that the Gaussian curvature can be defined almost everywhere on a good surface with non-positive total excess. Received April 3, 1997 / Published online February 5, 2001     

On conformal transformations of kropina metric

M. Hashiguchi [3] has studied the conformal theory of Finsler spaces. The theory of Kropina metric was investigated by L. Berwald [1] and V. K. Kropina [4]. The purpose of the present paper is to establish the conformal theory of Kropina metric. In this paper the transformation formulae for the difference tensor D _ik ⁱ (x, ) and Cartan's connection coefficients _k ^*i (x, ) have been obtained.     

New results on the proximal point algorithm in non-positive curvature metric spaces

The subject of this paper is the inexact proximal point algorithm of usual and Halpern type in non-positive curvature metric spaces. We study the convergence of the sequences given by the inexact proximal point algorithm with non-summable errors. We also prove the strong convergence of the Halpern proximal point algorithm to a minimum point of the convex function. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces.     

Simplicial nonpositive curvature

We introduce a family of conditions on a simplicial complex that we call local k-largeness (k≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that k-largeness implies non-positive curvature if k is sufficiently large. We also show that locally k-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.     

Bundle stabilisation and positive Ricci curvature

If E is the total space of a vector bundle over a compact Ricci non-negative manifold, it is known that E×Rp admits a complete metric of positive Ricci curvature for all sufficiently large p. In this paper we establish a small, explicit lower bound for the dimension p.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

Ricci curvature in the neighborhood of rank-one symmetric spaces

We study the Ricci curvature of a Riemannian metric as a differential operator acting on the space of metrics close (in a weighted functional spaces topology) to the standard metric of a rank-one noncompact symmetric space. We prove that any symmetric bilinear field close enough to the standard may be realized as the Ricci curvature of a unique close metric if its decay rate at infinity (its weight) belongs to some precisely known interval. We also study what happens if the decay rate is too small or too large.     

Examples of Hermitian manifolds with pointwise constant anti-holomorphic sectional curvature

We construct non-compact examples of Hermitian manifolds with pointwise constant anti-holomorphic sectional curvature. Our examples are obtained by conformal change of the metric on an open set of the complex space form.