We consider black brane spacetimes that have at least one spatial translation Killing field that is tangent to the brane. A new parameter, the tension of a spacetime, is defined. The tension parameter is associated with spatial translations in much the same way that the ADM mass is associated with the time translation Killing field. In this work, we explore the implications of the spatial translation symmetry for small perturbations around a background black brane. For static-charged black branes we derive a law which relates the tension perturbation to the surface gravity times the change in the horizon area, plus terms that involve variations in the charges and currents. We find that as a black brane evaporates the tension decreases. We also give a simple derivation of a first law for black brane spacetimes. These constructions hold when the background stress–energy is governed by a Hamiltonian, and the results include arbitrary perturbative stress–energy sources.
A contact path geometry is a family of paths in a contact manifold each of which is everywhere tangent to the contact distribution and such that given a point and a one-dimensional subspace of the contact distribution at that point there is a unique path of the family passing through the given point and tangent to the given subspace. A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifold and using this the local equivalence problem for contact projective structures is solved by the construction of a canonical regular Cartan connection. This Cartan connection is normal if and only if an invariant contact torsion vanishes. Every contact projective structure determines canonical paths transverse to the contact structure which fill out the contact projective structure to give a full projective structure, and the vanishing of the contact torsion implies the contact projective ambient connection agrees with the Thomas ambient connection of the corresponding projective structure. An analogue of the classical Beltrami theorem is proved for pseudo-hermitian manifolds with transverse symmetry.
This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical Kähler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an n-dimensional cone a rescaling of the canonical potential is an n-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Ampère metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-Kähler space.The ∞-Ricci tensor was introduced by A. Lichnerowicz in [46] where he used it to prove a generalization of the Cheeger-Gromoll splitting theorem.The idea related to these tensors relevant in what follows is that a lower bound on the (N + n)dimensional Bakry Emery Ricci tensor of the metric measure structure (k, φ) on the n-manifold M corresponds to a lower bound on the usual Ricci curvature of some metric associated to (k, φ) on some (n + N )-dimensional manifold fibering over M . In the example relevant here, N = n and the fibering manifold is the tube domain over an n-dimensional convex cone.One aim of this paper is to explain that it is profitable to regard a Riemannian Hessian metric g ij = ∇ i dF j as the mm-space determined by g ij in conjunction with the volume form H(F )Ψ = H(F ) 1/2 vol g , which is the pullback via the differential dF of the parallel volume form dual to Ψ on the dual affine space R n+1 * . More generally, a Kähler affine structure (g, ∇) is identified with the local mm-structure formed by g and the half Koszul one-form 1 2 H i .
An affine hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetric. AH structures simultaneously generalize Weyl structures and abstract the geometric structure determined on a non-degenerate co-oriented hypersurface in flat affine space by its second fundamental form together with either the projective structure induced by the affine normal or that induced by the conormal Gauß map. There are proposed notions of Einstein equations for AH structures which for Weyl structures specialize to the usual Einstein Weyl equations and such that the AH structure induced on a non-degenerate co-oriented affine hypersurface is Einstein if and only if the hypersurface is an affine hypersphere. It is shown that a convex flat projective structure admits a metric with which it generates an Einstein AH structure, and examples are constructed on mean curvature zero Lagrangian submanifolds of certain para-Kähler manifolds. The rough classification of Riemannian Einstein Weyl structures by properties of the scalar curvature is extended to this setting. Known estimates on the growth of the cubic form of an affine hypersphere are partly generalized. The Riemannian Einstein equations are reformulated in terms of a given background metric as an algebraically constrained elliptic system for a cubic tensor. From certain commutative nonassociative algebras there are constructed examples of exact Riemannian signature Einstein AH structures with self-conjugate curvature but which are not Weyl and are neither projectively nor conjugate projectively flat. 65 6. Riemannian Einstein AH structures with self-conjugate curvature 68 6.1. Symmetric tensor algebra 68 6.2. Differential operators on trace-free symmetric tensors 70 6.3. Curvature operator on trace-free symmetric tensors 73 6.4. Weitzenböck type identities for trace-free symmetric tensors 76 6.5. Alternative formulations of the Riemannian Einstein AH equations 80 6.6. Growth of the cubic torsion 85 6.7. Infinitesimal automorphisms of convex flat projective structures 91 7. Einstein AH structures from commutative nonassociative algebras 95 7.1. Partial differential equations for homogeneous cubic polynomials 95 7.2. Einstein commutative Codazzi algebras 101 8. The Einstein AH structure on a mean curvature zero Lagrangian submanifold of a para-Kähler space form 113 9. Analogy with conformal structures 119 9.1. Möbius structures, the Codazzi Laplacian, and an analogue of the Yamabe problem 120 9.2. Generalizing the Bach tensor 123 References 125
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