Integral formulas in riemannian geometry
01 Mar 1973-Bulletin of The London Mathematical Society (Oxford University Press (OUP))-Vol. 5, Iss: 1, pp 124-125
About: This article is published in Bulletin of The London Mathematical Society.The article was published on 1973-03-01. It has received 331 citations till now. The article focuses on the topics: Riemannian geometry & Fundamental theorem of Riemannian geometry.
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TL;DR: In this article, the exterior Einstein equations are explored from a differential geometric point of view using methods of global analysis and infinite-dimensional geometry, and the relationship between solutions with different shifts by finding the flow of a time-dependent vector field is clarified.
Abstract: In this paper the exterior Einstein equations are explored from a differential geometric point of view. Using methods of global analysis and infinite-dimensional geometry, we answer sharply the question: "In what sense are the Einstein equations, written as equations of evolution, a Lagrangian dynamical system?" By using our global methods, several aspects of the lapse function and shift vector field are clarified. The geometrical significance of the shift becomes apparent when the Einstein evolution equations are written using Lie derivatives. The evolution equations are then interpreted as evolution equations as seen by an observer in space coordinates. Using the notion of body-space transitions, we then find the relationship between solutions with different shifts by finding the flow of a time-dependent vector field. The use of body and space coordinates is shown to be somewhat analogous to the use of such coordinates in Euler's equations for a rigid body and the use of Eulerian and Lagrangian coordinates in hydrodynamics. We also explore the geometry of the lapse function, and show how one can pass from one lapse function to another by integrating ordinary differential equations. This involves integrating what we call the "intrinsic shift vector field." The essence of our method is to extend the usual configuration space [fraktur M]=Riem(M) of Riemannian metrics to [script T]×[script D]×[fraktur M], where [script T]=C[infinity](M,R) is the group of relativistic time translations and [script D]=Diff(M) is the group of spatial coordinate transformations of M. The lapse and shift then enter the dynamical picture naturally as the velocities canonically conjugate to the configuration fields (xit,etat)[is-an-element-of][script T]×[script D]. On this extended configuration space, a degenerate Lagrangian system is constructed which allows precisely for the arbitrary specification of the lapse and shift functions. We reinterpret a metric given by DeWitt for [fraktur M] as a degenerate metric on [script D]×[fraktur M]. On [script D]×[fraktur M], however, the metric is quadratic in the velocity variables. The groups [script T] and [script D] also serve as symmetry groups for our dynamical system. We establish that the associated conserved quantities are just the usual "constraint equations." A precise theorem is given for a remark of Misner that in an empty space-time we must have [script H]=0. We study the relationship between the evolution equations for the time-dependent metric gt and the Ricci flat condition of the reconstructed Lorentz metric gL. Finally, we make some remarks about a possible "superphase space" for general relativity and how our treatment on [script T]×[script D]×[fraktur M] is related to ordinary superspace and superphase space.
97 citations
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TL;DR: The eigenvalues and degeneracies of the covariant Laplacian acting on symmetric tensors of rank m≤2 defined on n‐spheres with n≥3 are given.
Abstract: The eigenvalues and degeneracies of the covariant Laplacian acting on symmetric tensors of rank m≤2 defined on n‐spheres with n≥3 are given.
96 citations
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TL;DR: In this article, it was shown that the quasi-Euclidean sections of various rotating black holes in different dimensions possess at least one nonconformal negative mode when thermodynamic instabilities are expected.
Abstract: We show that the quasi-Euclidean sections of various rotating black holes in different dimensions possess at least one nonconformal negative mode when thermodynamic instabilities are expected. The boundary conditions of the fixed induced metric correspond to the partition function of the grand-canonical ensemble. Indeed, in the asymptotically flat cases, we find that a negative mode persists even if the specific heat at constant angular momenta is positive, since the stability in this ensemble also requires the positivity of the isothermal moment of inertia. We focus, in particular, on Kerr black holes, on Myers-Perry black holes in five and six dimensions, and on the Emparan-Reall black ring solution. We go on further to consider the richer case of the asymptotically AdS Kerr black hole in four dimensions, where thermodynamic stability is expected for a large enough cosmological constant. The results are consistent with previous findings in the nonrotation limit and support the use of quasi-Euclidean instantons to construct gravitational partition functions.
92 citations
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TL;DR: In this article, the authors define almost Yamabe solitons as special conformal solutions of the Yamabe flow and obtain some rigidity results concerning Yamabe almost-solitons.
Abstract: The aim of this note is to define almost Yamabe solitons as special conformal solutions of the Yamabe flow. Moreover, we shall obtain some rigidity results concerning Yamabe almost solitons. Finally, we shall give some characterizations for homogeneous gradient Yamabe almost solitons.
81 citations
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TL;DR: In this article, it was shown that any compact non-trivial almost Ricci soliton with constant scalar curvature is isometric to a Euclidean sphere.
Abstract: The aim of this note is to prove that any compact non-trivial almost Ricci soliton $$\big (M^n,\,g,\,X,\,\lambda \big )$$
with constant scalar curvature is isometric to a Euclidean sphere $$\mathbb {S}^{n}$$
. As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field $$X$$
decomposes as the sum of a Killing vector field $$Y$$
and the gradient of a suitable function.
59 citations