Spacetime atoms and extrinsic curvature of equi-geodesic surfaces

The field equations associated to Born–Infeld type brane theories are studied by using auxiliary variables. This approach hinges on the fact, that the expressions defining the physical and geometrical quantities describing the worldvolume are varied independently. The general structure of the Born–Infeld type theories for branes contains the square root of a determinant of a combined matrix between the induced metric on the worldvolume swept out by the brane and a symmetric/antisymmetric tensor depending on gauge, matter or extrinsic curvature terms taking place on the worldvolume. The higher-order curvature terms appearing in the determinant form come to play in competition with other effective brane models. Additionally, we suggest a Born–Infeld–Einstein type action for branes where the higher-order curvature content is provided by the worldvolume Ricci tensor. This action provides an alternative description of the dynamics of braneworld scenarios.

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COVARIANT CONFORMAL DECOMPOSITION OF EINSTEIN EQUATIONS

International Journal of Modern Physics A ◽

10.1142/s0217751x02011898 ◽

2002 ◽

Vol 17 (20) ◽

pp. 2762-2762

Author(s):

E. GOURGOULHON ◽

J. NOVAK

Keyword(s):

Numerical Simulations ◽

Extrinsic Curvature ◽

Einstein Equations ◽

Numerical Implementation ◽

Einstein’S Equations ◽

Einstein's Equations ◽

Tensor Density ◽

Dirac Gauge ◽

Ill Posed ◽

Isolated Systems

It has been shown1,2 that the usual 3+1 form of Einstein's equations may be ill-posed. This result has been previously observed in numerical simulations3,4. We present a 3+1 type formalism inspired by these works to decompose Einstein's equations. This decomposition is motivated by the aim of stable numerical implementation and resolution of the equations. We introduce the conformal 3-"metric" (scaled by the determinant of the usual 3-metric) which is a tensor density of weight -2/3. The Einstein equations are then derived in terms of this "metric", of the conformal extrinsic curvature and in terms of the associated derivative. We also introduce a flat 3-metric (the asymptotic metric for isolated systems) and the associated derivative. Finally, the generalized Dirac gauge (introduced by Smarr and York5) is used in this formalism and some examples of formulation of Einstein's equations are shown.

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ON EIGENVALUE ESTIMATES FOR THE SUBMANIFOLD DIRAC OPERATOR

International Journal of Mathematics ◽

10.1142/s0129167x0200140x ◽

2002 ◽

Vol 13 (05) ◽

pp. 533-548 ◽

Cited By ~ 11

Author(s):

NICOLAS GINOUX ◽

BERTRAND MOREL

Keyword(s):

Dirac Operator ◽

Lower Bounds ◽

Dirac Operators ◽

Extrinsic Curvature ◽

Eigenvalue Estimates ◽

Killing Spinors ◽

Spinor Fields ◽

Twisted Dirac Operators

We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class of spinor fields generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

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Totally geodesic surfaces and homology

Algebraic & Geometric Topology ◽

10.2140/agt.2006.6.1413 ◽

2006 ◽

Vol 6 (3) ◽

pp. 1413-1428 ◽

Cited By ~ 1

Author(s):

Jason DeBlois

Keyword(s):

Totally Geodesic ◽

Geodesic Surfaces

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Anomalies generated by extrinsic curvature

Zeitschrift für Physik C Particles and Fields ◽

10.1007/bf01573944 ◽

1987 ◽

Vol 36 (3) ◽

pp. 479-486 ◽

Cited By ~ 3

Author(s):

F. Langouche ◽

H. Leutwyler

Keyword(s):

Extrinsic Curvature

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Manifold curvature and Ehrenfest forces with a moving basis

SciPost Physics ◽

10.21468/scipostphys.12.1.020 ◽

2022 ◽

Vol 12 (1) ◽

Author(s):

Jessica Halliday ◽

Emilio Artacho

Keyword(s):

Extrinsic Curvature ◽

Quantum States ◽

Basis Set ◽

Electronic Subsystem ◽

Intrinsic Curvature ◽

Geometrical Meaning

Known force terms arising in the Ehrenfest dynamics of quantum electrons and classical nuclei, due to a moving basis set for the former, can be understood in terms of the curvature of the manifold hosting the quantum states of the electronic subsystem. Namely, the velocity-dependent terms appearing in the Ehrenfest forces on the nuclei acquire a geometrical meaning in terms of the intrinsic curvature of the manifold, while Pulay terms relate to its extrinsic curvature.

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Non-simple geodesics in hyperbolic 3-manifolds

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100072625 ◽

1994 ◽

Vol 116 (2) ◽

pp. 339-351

Author(s):

Kerry N. Jones ◽

Alan W. Reid

Keyword(s):

Closed Geodesic ◽

Closed Geodesics ◽

Totally Geodesic ◽

Geodesic Surfaces ◽

Simple Closed Geodesics

AbstractChinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

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