Krishnan Shankar

Rank two fundamental groups of positively curved manifolds

This paper deals with the construction of previously unknown fundamental groups for positively curved manifolds.

Isometry groups of homogeneous spaces with positive sectional curvature

Abstract We calculate the full isometry group in the case G/H admits a homogeneous metric of positive sectional curvature.

Spherical rank rigidity and Blaschke manifolds

Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=\pi. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank, then M is isometric to a compact, rank one symmetric space (CROSS) i.e., isometric to a sphere, complex projective space, quaternionic projective space or to the Cayley plane. The notion of spherical rank is analogous to the notions of Euclidean rank and hyperbolic rank studied by several people (see references). The main theorem is proved in two steps: first we show that M is a so called Blaschke manifold with extremal injectivity radius (equal to diameter). Then we prove that such M is isometric to a CROSS.

Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive rational number r, we associate a finite, aspherical 2-complex X_{r,P} and calculate the Dehn function of its fundamental group G_{r,P} in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes x^s, where s is an arbitrary rational number greater than or equal to 2. By repeatedly forming multiple HNN extensions of the groups G_{r,P} we exhibit a similar range of behavior among higher-dimensional Dehn functions, proving in particular that for each positive integer k and rational s greater than or equal to (k+1)/k there exists a group with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are obtained for arbitrary manifold pairs (M,\partial M) in addition to (B^{k+1},S^k).

On complexes equivalent to S3-bundles over S4

S3-bundles over S4 have played an important role in topology and geometry since Milnor showed that the total spaces of such bundles with Euler class ±1 are manifolds homeomorphic to S7 but not always diffeomorphic to it. In 1974, Gromoll and Meyer exhibited one of these spheres (a generator in the group of homotopy 7-spheres) as a double coset manifold i.e. a quotient of Sp(2) hence showing that it admits a metric of nonnegative curvature (cf. [6]). Until recently, this was the only exotic sphere known to admit a metric of nonnegative sectional curvature. Then in [7], K. Grove and W. Ziller constructed metrics of nonnegative curvature on the total space of S3-bundles over S4. They also asked for a classification of these bundles up to homotopy equivalence, homeomorphism and diffeomorphism. These questions have been addressed in many papers such as [12], [11], [15] and more recently in [3]. In this paper we attempt to fill the gap in the previous papers; we consider the problem of determining when a given CW complex is homotopy equivalent to such a bundle. The problem was motivated by [7]: the Berger space, Sp(2)/Sp(1), is a 7-manifold that has the cohomology ring of an S3-bundle over S4, but does it admit the structure of such a bundle? The fact that it cannot be a principal S3-bundle over S4 is straightforward and is proved in [7].

Positively curved manifolds with large spherical rank

Rigidity results are obtained for Riemannian d-manifolds with sec > 1 and spherical rank at least d − 2 > 0. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini– Study metric. This conjecture is verified in all odd dimensions, for metrics on dspheres when d 6= 6, for Riemannian manifolds satisfying the Rakic duality principle, and for Kahlerian manifolds.

Nonnegatively and positively curved invariant metrics on circle bundles

We derive and study necessary and sufficient conditions for an S 1 -bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection metric of nonnegative curvature on the trivial .bundle S 1 x S 3 that is not a product metric.