Closed geodesics in stationary manifolds with strictly convex boundary

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.

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Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004118000543 ◽

2018 ◽

Vol 168 (1) ◽

pp. 29-41 ◽

Cited By ~ 1

Author(s):

JOONAS ILMAVIRTA ◽

JERE LEHTONEN ◽

MIKKO SALO

Keyword(s):

Higher Dimensions ◽

Uniqueness Result ◽

Constant Function ◽

Piecewise Constant Function ◽

Two Dimensional ◽

Local Uniqueness ◽

Piecewise Constant ◽

Convex Boundary ◽

X Ray ◽

Strictly Convex

AbstractWe show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

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Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

Journal of Geometric Analysis ◽

10.1007/s12220-017-9861-9 ◽

2017 ◽

Vol 28 (2) ◽

pp. 1245-1257

Author(s):

Abraão Mendes

Keyword(s):

Free Boundary ◽

Convex Boundary ◽

Strictly Convex ◽

Boundary Surfaces

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Marked boundary rigidity for surfaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.94 ◽

2016 ◽

Vol 38 (4) ◽

pp. 1459-1478 ◽

Cited By ~ 2

Author(s):

COLIN GUILLARMOU ◽

MARCO MAZZUCCHELLI

Keyword(s):

Riemannian Metrics ◽

Compact Surface ◽

Conjugate Points ◽

Geodesic Flows ◽

Convex Boundary ◽

Strictly Convex ◽

Negatively Curved ◽

Boundary Rigidity ◽

Boundary Distance

We show that, on an oriented compact surface, two sufficiently $C^{2}$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows and the same marked boundary distance are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and the same marked boundary distance, extending a result of Croke and Otal.

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Filling words in fundamental groups of Riemann surfaces

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.50.2013.1.1227 ◽

2013 ◽

Vol 50 (1) ◽

pp. 31-50

Author(s):

C. Zhang

Keyword(s):

Riemann Surface ◽

Fundamental Group ◽

Riemann Surfaces ◽

Fundamental Groups ◽

Closed Geodesics ◽

New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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