scholarly journals Stitching Data: Recovering a Manifold’s Geometry from Geodesic Intersections

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Reed Meyerson
Keyword(s):  
Riemannian Manifold ◽  
Measurement Problem ◽  
Manifold With Boundary ◽  
Boundary Measurement ◽  
Data Problem

AbstractLet (M, g) be a Riemannian manifold with boundary. We show that knowledge of the length of each geodesic, and where pairwise intersections occur along the corresponding geodesics allows for recovery of the geometry of (M, g) (assuming (M, g) admits a Riemannian collar of a uniform radius). We call this knowledge the ‘stitching data’. We then pose a boundary measurement problem called the ‘delayed collision data problem’ and apply our result about the stitching data to recover the geometry from the collision data (with some reasonable geometric restrictions on the manifold).

scholarly journals Boundary Value Problems for the Lorentzian Dirac Operator

2018 ◽  
pp. 3-18
Author(s):  
Christian Bär ◽  
Sebastian Hannes
Keyword(s):  
Riemannian Manifold ◽  
Dirac Operator ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Classical Case ◽  
Spin Manifold ◽  
Globally Hyperbolic ◽  
Manifold With Boundary ◽  
Smooth Space

On a compact globally hyperbolic Lorentzian spin manifold with smooth space-like Cauchy boundary, the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah–Patodi–Singer boundary conditions are imposed. This chapter explores to what extent these boundary conditions can be replaced by more general ones and how the index then changes. There are some differences to the classical case of the elliptic Dirac operator on a Riemannian manifold with boundary.

scholarly journals A long neck principle for Riemannian spin manifolds with positive scalar curvature

2020 ◽  
Vol 30 (5) ◽  
pp. 1183-1223
Author(s):  
Simone Cecchini
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Pairwise Disjoint ◽  
Index Theory ◽  
Spin Manifold ◽  
Manifolds With Boundary ◽  
Topological Information ◽  
Manifold With Boundary ◽  
Spin Manifolds ◽  
Nonzero Degree

AbstractWe develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a “long neck principle” for a compact Riemannian spin n-manifold with boundary X, stating that if $${{\,\mathrm{scal}\,}}(X)\ge n(n-1)$$ scal ( X ) ≥ n ( n - 1 ) and there is a nonzero degree map into the sphere $$f:X\rightarrow S^n$$ f : X → S n which is strictly area decreasing, then the distance between the support of $$\text {d}f$$ d f and the boundary of X is at most $$\pi /n$$ π / n . This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold X obtained by removing k pairwise disjoint embedded n-balls from a closed spin n-manifold Y. We show that if $${{\,\mathrm{scal}\,}}(X)>\sigma >0$$ scal ( X ) > σ > 0 and Y satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $$\partial X$$ ∂ X is at most $$\pi \sqrt{(n-1)/(n\sigma )}$$ π ( n - 1 ) / ( n σ ) . Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to $$N\times [-1,1]$$ N × [ - 1 , 1 ] , with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if $${{\,\mathrm{scal}\,}}(V)\ge \sigma >0$$ scal ( V ) ≥ σ > 0 , then the distance between the boundary components of V is at most $$2\pi \sqrt{(n-1)/(n\sigma )}$$ 2 π ( n - 1 ) / ( n σ ) . This last constant is sharp by an argument due to Gromov.

Existence of a Closed Geodesic on Non-compact Riemannian Manifolds with Boundary

2002 ◽  
Vol 2 (1) ◽  
Author(s):  
Rossella Bartolo ◽  
Anna Germinario ◽  
Miguel Sánchez
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Closed Geodesic ◽  
Detailed Comparison ◽  
Manifolds With Boundary ◽  
Manifold With Boundary

AbstractA new result about the existence of a closed geodesic on a Riemannian manifold with boundary is given. A detailed comparison with previous results is carried out.

scholarly journals Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary

2017 ◽  
Vol 8 (1) ◽  
pp. 559-582 ◽  
Author(s):  
Mónica Clapp ◽  
Marco Ghimenti ◽  
Anna Maria Micheletti
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Compact Riemannian Manifold ◽  
Semiclassical Limit ◽  
Arbitrary Dimension ◽  
Neumann Boundary ◽  
Perturbation Parameter ◽  
Singularly Perturbed ◽  
Manifold With Boundary ◽  
Klein Gordon

Abstract We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold {\mathfrak{M}} with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of {\mathfrak{M}} , forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in {\mathbb{R}^{N}} . Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

THE SKELETON OF A RIEMANNIAN MANIFOLD WITH BOUNDARY

1978 ◽  
Vol 33 (3) ◽  
pp. 181-182 ◽  
Author(s):  
A G Vainshtein ◽  
V A Efremovich ◽  
E A Loginov
Keyword(s):  
Riemannian Manifold ◽  
Manifold With Boundary
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