Projective Lichnerowicz-Obata Conjecture

An Intrinsic Approach to Lichnerowicz Conjecture

In this paper we give a proof of Lichnerowicz Conjecture for compact simply connected manifolds which is intrinsic in the sense that it avoids the Nice Embeddings into eigen spaces of the Laplacian. Even if one wants to use these embeddings this paper gives a more streamlined proof.

Lichnerowicz and Obata Theorems for Foliations

The standard Lichnerowicz comparison theorem states that if the Ricci curvature of a closed, Riemannian n-manifold M satisfies Ric (X,X) ≥ a (n − 1) |X| for every X ∈ TM for some fixed a > 0, then the smallest positive eigenvalue λ of the Laplacian satisfies λ ≥ an. The Obata theorem states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvatu...

The Auslander-Reiten Conjecture for Group Rings

This paper studies the vanishing of $Ext$ modules over group rings. Let $R$ be a commutative noetherian ring and $ga$ a group. We provide a criterion under which the vanishing of self extensions of a finitely generated $Rga$-module $M$ forces it to be projective. Using this result, it is shown that $Rga$ satisfies the Auslander-Reiten conjecture, whenever $R$ has finite global dimension and $ga...

A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler manifold whose Ricci curvature satisfies Ric ≥ k for some constant k > 0. When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field. We show that a ball of sufficiently large radius in complex projective space provides an example of a s...

Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz