Isometry-invariant geodesics and the fundamental group, II

Isometry-invariant geodesics and the fundamental group

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true if the fundamental group is infinite cyclic. We also formulate a generalization of the isometry-invariant geodesics problem, and a generalization of the cele...

Isometry - invariant geodesics with Lipschitz obstacle 1

Given a linear isometry A0 : Rn → Rn of finite order on Rn , a general 〈A0〉-invariant closed subset M of Rn is considered with Lipschitz boundary. Under suitable topological restrictions the existence of A0-invariant geodesics of M is proven.

Isometry-invariant Geodesics and Nonpositive Derivations of the Cohomology

We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of Q-cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M with H(M, Q) belonging to this class, every isometry has a non-trivial invariant geodesic, for any metric on M . We use Qsurgery to construc...

Group Invariant Integration and the Fundamental Theorem of Algebra.

The fundamental group is not a derived invariant

We show that the fundamental group is not invariant under derived equivalence of smooth projective varieties. 2010 Mathematics Subject Classification. Primary 14-F05; Secondary 14-F35.