Christopher B. Croke

Spaces with nonpositive curvature and their ideal boundaries

Abstract We construct a pair of finite piecewise Euclidean 2-complexes with nonpositive curvature which are homeomorphic but whose universal covers have nonhomeomorphic ideal boundaries, settling a question of Gromov.

Spectral rigidity of a compact negatively curved manifold

A non-blinking rotary electromagnetic indicator having an annular stator with equiangularly spaced radially inwardly extending salient poles with windings disposed thereon and a rotor with a permanent magnet rotatable into alignment with the salient poles in accordance with the operation of the stator windings and a symbol bearing drum encircling the stator. A rotor vane of ferromagnetic material is provided next to each pole of the rotor magnet angularly offset from the magnetic axis of the permanent magnet and extending outwardly of the permanent magnet pole to provide a turning force on the rotor when it is to be rotated to a 180 DEG opposite position.

The marked length-spectrum of a surface of nonpositive curvature☆

We generalize work of J.P. Otal and C. Croke on the marked length spectrum of surfaces to the case where one of the metrics is of nonpositive curvature and the other one has no conjugate points. If M is a manifold and g1, g2 are two Riemannian metrics, we say that they have the same marked length spectrum if in each homotopy class of closed curves in M the infimum of g1-lengths of curves and the infimum of g2-lengths of curves are the same. The marked length spectrum problem in general is to show that two metrics with the same marked length spectrum are isometric. Of course, this cannot hold for arbitrary metrics (for example if M is simply connected). This problem was stated as a conjecture in [BK] in the case where M is a closed surface and g1 and g2 are of negative curvature. This conjecture was solved by J.P. Otal [To] and independently by C. Croke [Cr]–see also [Fa]. Previous work on the problem was done by Guillemin and Kazhdan [GK]. In this work, using Otal’s approach, we improve some of these results by proving the following theorem: Theorem A. Let M be a closed surface and let g1, g2 be Riemannian metrics on M , with g1 of nonpositive curvature and g2 without conjugate points. If g1 and g2 have the same marked length-spectrum then they are isometric by an isometry homotopic to the identity. We will also prove the following fact, which reduces the length spectrum and curvature condition to the assumption that the Morse correspondence preserves angles—see §1 for the definition of the Morse correspondence. Theorem B. Let M be a closed surface of genus ≥ 2, and let g1, g2 be Riemannian metrics without conjugate points on M . If g1 and g2 have the same marked lengthspectrum and the Morse correspondence preserves angles then they are isometric by an isometry homotopic to the identity.

An eigenvalue pinching theorem

In 1958 Lichnerowicz [7] showed that for a compact n-dimensional riemannian manifold M, whose Ricci curvature is bounded below by n 1 , the first non-zero eigenvalue, 21, of the laplacian satisfies 2 t>n . If, in fact, 21 =n Obata proved that M must be isometric to the standard sphere. A natural question is: Do there exist constants C(n)> 1, depending only on n such that if M is as above and C(n). n> 21 > n then M must be diffeomorphic to a sphere. Here, by combining the works of Gromov [3], Berard and Meyer [1], and Grove and Shiohama [4], we show:

Boundary case of equality in optimal Loewner-type inequalities

We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, G), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T b , which are area-decreasing (on b-dimensional areas), with respect to suitable norms. These norms are the stable norm of G, the conformally invariant norm, as well as other L p -norms. Here we exploit L P -minimizing differential 1-forms in cohomology classes. We characterize the case of equality in our optimal inequalities, in terms of the criticality of the lattice of deck transformations of T b , while the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, under the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the Abel-Jacobi map to the maximal free abelian cover.

The fundamental group of compact manifolds without conjugate points

The fundamental group of compact manifolds without conjugate points.

RIGIDITY THEOREMS IN RIEMANNIAN GEOMETRY

The purpose of this chapter is to survey some recent results and state open questions concerning the rigidity of Riemannian manifolds. The starting point will be the boundary rigidity and conjugacy rigidity problems. These problems are connected to many other problems (Mostow-Margulis type rigidity, isopectral problems, isoperimetric inequalities etc.). We will restrict our attention to those results that have a direct connection to the boundary rigidity problem (see Section 2) or the conjugacy rigidity problem (see Section 4). Even with that restriction the connections are numerous and the author was forced to select the topics covered here in accordance with rather subjective criteria. A few of the topics not covered are mentioned in Section 11 but there are others.

On tori without conjugate points

SummaryIn this paper we consider Riemannian metrics without conjugate points on an n-torus. Recent work of J. Heber established that the gradient vector fields of Busemann functions on the universal cover of such a manifold induce a natural foliation (akin to the weak stable foliation for a Riemannian manifold with negative sectional curvature) on the unit tangent bundle. The main result in the paper is that the metric is flat if this foliation is Lipschitz. We also prove that this foliation is Lipschitz if and only if the metric has bounded asymptotes. This confirms a conjecture of E. Hopf in this case.

Conjugacy and rigidity for nonpositively curved manifolds of higher rank

Abstract Let M and N be compact Riemannian manifolds with sectional curvature K ⩽ 0 such that M has dimension ⩾ 3 and rank ⩾ 2. If there exists a C 0 conjugacy F between the geodesic flows of the unit tangent bundles of M and N , then there exists an isometry G : M → N that induces the same isomorphism as F between the fundamental groups of M and N .

A warped product splitting theorem

The celebrated Cheeger-Gromoll splitting theorem (see [ChGr], and also [EsHe, Wu]) implies that a connected, complete Riemannian manifold having nonnegative Ricci curvature and two ends is isometric to a Riemannian product, where one factor is a line. There is a version of this theorem for compact, connected Riemannian manifolds M with two smooth boundary components. It states that if M has nonnegative Ricci curvature and two boundary components, both of which have nonnegative mean curvature with respect to their inward normals, then M is isometric to the Riemannian product of N1 with a closed interval (in particular N1 is isometric to N2). In this paper we will prove an analog of this result for manifolds with Ricci curvature ≥ −(n − 1). Since our argument parallels the proof of the splitting theorem for manifolds with nonnegative Ricci curvature, we present the proofs simultaneously by letting δ be either zero or one in what follows.