# Morse theory, closed geodesics, and the homology of free loop spaces

@article{Oancea2014MorseTC, title={Morse theory, closed geodesics, and the homology of free loop spaces}, author={Alexandru Oancea}, journal={arXiv: Differential Geometry}, year={2014} }

This is a survey paper on Morse theory and the existence problem for closed geodesics. The free loop space plays a central role, since closed geodesics are critical points of the energy functional. As such, they can be analyzed through variational methods. The topics that we discuss include: Riemannian background, the Lyusternik-Fet theorem, the Lyusternik-Schnirelmann principle of subordinated classes, the Gromoll-Meyer theorem, Bott's iteration of the index formulas, homological computations…

## 19 Citations

The cohomology of free loop spaces of homogeneous spaces

- Mathematics
- 2017

The free loops space ΛX of a space X has become an important object of study particularly in the case when X is a manifold. The study of free loop spaces is motivated in particular by two main…

Spherical complexities with applications to closed geodesics

- Mathematics
- 2019

We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the…

The Existence of Infinitely Many Geometrically Distinct Non-Constant Prime Closed Geodesics on Riemannian Manifolds

- Mathematics
- 2018

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show…

M ay 2 01 9 The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form S 2 n + 1 / Γ

- 2019

Let M = S/Γ, Γ is a finite group which acts freely and isometrically on the (2n + 1)-sphere and therefore M is diffeomorphic to a compact space form. In this paper, we first investigate Katok’s…

ROBUSTNESS OF TOPOLOGICAL ENTROPY FOR GEODESIC FLOWS

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In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the C topology. We establish several instances of entropy robustness…

$C^0$-Robustness of topological entropy for geodesic flows

- Mathematics
- 2021

In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the C topology. We establish several instances of entropy robustness…

Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

- Mathematics
- 2018

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance…

The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $$S^{2n+1}/ \Gamma $$S2n+1/Γ

- MathematicsCalculus of Variations and Partial Differential Equations
- 2019

Let $$M=S^{2n+1}/ \Gamma $$M=S2n+1/Γ, $$\Gamma $$Γ is a finite group which acts freely and isometrically on the $$(2n+1)$$(2n+1)-sphere and therefore M is diffeomorphic to a compact space form. In…

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

- Mathematics, Physics
- 2017

Let $M=S^n/ \Gamma$ and $h$ be a nontrivial element of finite order $p$ in $\pi_1(M)$, where the integer $n\geq2$, $\Gamma$ is a finite group which acts freely and isometrically on the $n$-sphere and…

The Existence of Infinitely Many Geometrically Distinct Prime Non-Constant Closed Geodesics on Riemannian Manifolds

- Physics
- 2018

We enumerate a necessary condition for the existence of infinitely many geometrically distinct prime non-constant closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show…

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