scholarly journals Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

2021 ◽  
Author(s):  
Bernd Ammann ◽  
Klaus Kröncke ◽  
Olaf Müller
Keyword(s):  
Riemannian Metrics ◽  
Mathematical Physics ◽  
Closed Curve ◽  
Data Sets ◽  
Lorentzian Manifolds ◽  
Constraint Equations ◽  
Parallel Spinors ◽  
Space Of Riemannian Metrics ◽  
Geometry Analysis ◽  
Parallel Spinor

AbstractLorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on M with a parallel spinor gives rise to a solution of the constraint equations on $$M\times (a,b)$$ M × ( a , b ) (resp. $$M\times S^1$$ M × S 1 ).

scholarly journals Rigidity and stability of Einstein metrics for quadratic curvature functionals

2015 ◽  
Vol 2015 (700) ◽  
Author(s):  
Matthew J. Gursky ◽  
Jeff A. Viaclovsky
Keyword(s):  
Critical Points ◽  
Moduli Space ◽  
Einstein Metrics ◽  
Riemannian Metrics ◽  
Local Minimizers ◽  
Lagrange Equations ◽  
Critical Metrics ◽  
Local Properties ◽  
Space Of Riemannian Metrics ◽  
Curvature Functionals

AbstractWe investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of Riemannian metrics. We show it is possible to “gauge” the Euler–Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. We show a number of compact examples are infinitesimally rigid, and consequently, are isolated critical points in the space of unit-volume Riemannian metrics. We then give examples of critical metrics which are strict local minimizers (up to diffeomorphism and scaling). A corollary is a local “reverse Bishop's inequality” for such metrics. In particular, any metric

ITERATIVE GEOMETRIC STRUCTURES

2010 ◽  
Vol 07 (07) ◽  
pp. 1103-1114 ◽  
Author(s):  
GABRIEL BERCU ◽  
CLAUDIU CORCODEL ◽  
MIHAI POSTOLACHE
Keyword(s):  
Differential Geometry ◽  
Special Functions ◽  
Riemannian Metrics ◽  
General Setting ◽  
Mathematical Physics ◽  
Geometric Structures ◽  
Geometric Characteristics ◽  
Geometrical Characteristics ◽  
Metric Connection

In this work, we propose a study of geometric structures (connections, pseudo-Riemannian metrics) adapted to some fundamental problems of Differential Geometry. Then we find geometrical characteristics of some ODE or PDE of Mathematical Physics. While Sec. 1 contains the general setting, Secs. 2–5 contain our results. In Sec. 2, we introduce a Hessian structure having the same connection as the initial metric. In Sec. 3, we initiate a study on iterative 2D Hessian structures. In Sec. 4, we find pairs (metric, connection) generated by special functions. In Sec. 5, we find geometric characteristics of a PDE.

scholarly journals Computability Theory and Differential Geometry

2004 ◽  
Vol 10 (4) ◽  
pp. 457-486 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Turing Machine ◽  
Riemannian Metrics ◽  
Computable Function ◽  
Computability Theory ◽  
Settling Time ◽  
Point Of View ◽  
Connected Components ◽  
Local Minima ◽  
Smooth Compact Manifold ◽  
Space Of Riemannian Metrics

Abstract. LetMbe a smooth, compact manifold of dimensionn≥ 5 and sectional curvature ∣K∣ ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics onMmodulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machineTe,eϵ ω, there is a sequence (uniformly effective ine) of homologyn-sphereswhich are also hypersurfaces, such thatis diffeomorphic to the standardn-sphereSn(denoted)iffTehalts on inputk, and in this case the connected sum, so, andis associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time ae σe(k)ofTeon inputsy<k.At their request Soare constructed a particular infinite sequence {Ai}ϵωof c.e. sets so that for allithe settling time of the associated Turing machine forAidominates that forAi+1, even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a “fractal” like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as “the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization.” From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structuralcomplexityof the geometry and topology, not merelyundecidabilityresults as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they usenontrivialinformation about c.e. sets, the Soare sequence {Ai}iϵωabove, not merely Gödel's c.e. noncomputable set K of the 1930's; and (3)withoutusing computability theory there is no known proof that local minima exist even for simple manifolds like the torusT5(see §9.5).

scholarly journals The Hodge conjecture: The complications of understanding the shape of geometric spaces

2018 ◽  
pp. 51
Author(s):  
Vicente Muñoz Velázquez
Keyword(s):  
Differential Geometry ◽  
Algebraic Geometry ◽  
Natural Condition ◽  
Mathematical Physics ◽  
Complex Numbers ◽  
Hodge Conjecture ◽  
Real Numbers ◽  
Geometric Objects ◽  
Complex Submanifolds ◽  
Geometry Analysis

The Hodge conjecture is one of the seven millennium problems, and is framed within differential geometry and algebraic geometry. It was proposed by William Hodge in 1950 and is currently a stimulus for the development of several theories based on geometry, analysis, and mathematical physics. It proposes a natural condition for the existence of complex submanifolds within a complex manifold. Manifolds are the spaces in which geometric objects can be considered. In complex manifolds, the structure of the space is based on complex numbers, instead of the most intuitive structure of geometry, based on real numbers.

scholarly journals Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

2020 ◽  
Vol 63 (4) ◽  
pp. 901-908
Author(s):  
Philipp Reiser
Keyword(s):  
Scalar Curvature ◽  
Moduli Spaces ◽  
Riemannian Metrics ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Positive Scalar ◽  
Space Of Riemannian Metrics ◽  
Simply Connected

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

scholarly journals Gaussian measures on the of space of Riemannian metrics

2015 ◽  
Vol 39 (2) ◽  
pp. 129-145
Author(s):  
Brian Clarke ◽  
Dmitry Jakobson ◽  
Niky Kamran ◽  
Lior Silberman ◽  
Jonathan Taylor
Keyword(s):  
Riemannian Metrics ◽  
Gaussian Measures ◽  
Space Of Riemannian Metrics
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