About Curvature, Conformal Metrics and Warped Products
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Arxiv:Math/0406039V3
CURVATURE OF MULTIPLY WARPED PRODUCTS FERNANDO DOBARRO AND BULENT¨ UNAL¨ Abstract. In this paper, we study Ricci-flat and Einstein Lorentzian multiply warped products. We also consider the case of having constant scalar curvatures for this class of warped products. Finally, after we introduce a new class of space- times called as generalized Kasner space-times, we apply our results to this kind of space-times as well as other relativistic space-times, i.e., Reissner-Nordstr¨om, Kasner space-times, Ba˜nados-Teitelboim-Zanelli and de Sitter Black hole solutions. Contents 1. Introduction 2. Preliminaries 3. Special Multiply Warped Products 3.1. Einstein Ricci Tensor 3.2. Constant Scalar Curvature 4. Generalized Kasner Space-time 5. 4-Dimensional Space-time Models 5.1. Type (I) 5.2. Type (II) 5.3. Type (III) 6. BTZ (2 + 1)-Black Hole Solutions 7. Conclusions References arXiv:math/0406039v3 [math.DG] 11 Dec 2004 Date: October 26, 2004. 1991 Mathematics Subject Classification. 53C25, 53C50. Key words and phrases. Warped products, Ricci tensor, scalar curvature, Einstein manifolds. Research of F. D. partially supported by funds of the National Group ’Analisi Reale’ of the Italian Ministry of University and Scientific Research at the University of Trieste. 1 2 FERNANDODOBARROANDBULENT¨ UNAL¨ 1. Introduction The concept of warped products was first introduced by Bishop and O’Neill (see [20]) to construct examples of Riemannian manifolds with negative curvature. In Riemannian geometry, warped product manifolds and their generic forms have been used to construct new examples with interesting curvature properties since then (see [19, 20, 24, 27, 32, 40, 41, 52, 53, 54, 57, 60]). -
Arxiv:Math/0412436V4 [Math.DG] 29 Apr 2008 Eodr:3Q5 38,8E5 83E30
CURVATURE IN SPECIAL BASE CONFORMAL WARPED PRODUCTS FERNANDO DOBARRO & BULENT¨ UNAL¨ Abstract. We introduce the concept of a base conformal warped prod- uct of two pseudo-Riemannian manifolds. We also define a subclass of this structure called as a special base conformal warped product. After, we explicitly mention many of the relevant fields where metrics of these forms and also considerations about their curvature related properties play important rolls. Among others, we cite general relativity, extra- dimension, string and super-gravity theories as physical subjects and also the study of the spectrum of Laplace-Beltrami operators on p-forms in global analysis. Then, we give expressions for the Ricci tensor and scalar curvature of a base conformal warped product in terms of Ricci tensors and scalar curvatures of its base and fiber, respectively. Fur- thermore, we introduce specific identities verified by particular families of, either scalar or tensorial, nonlinear differential operators on pseudo- Riemannian manifolds. The latter allow us to obtain new interesting expressions for the Ricci tensor and scalar curvature of a special base conformal warped product and it turns out that not only the expressions but also the analytical approach used are interesting from the physical, geometrical and analytical point of view. Finally, we analyze, investigate and characterize possible solutions for the conformal and warping factors of a special base conformal warped product, which guarantee that the corresponding product is Einstein. Besides all, we apply these results to a generalization of the Schwarzschild metric. arXiv:math/0412436v4 [math.DG] 29 Apr 2008 Date: April 25, 2008. 1991 Mathematics Subject Classification. -
Notes on a Conformal Characterization of 2-Dimensional Lorentzian Manifolds with Constant Ricci Scalar Curvature
U.P.B. Sci. Bull., Series A, Vol. 83, Iss. 2, 2021 ISSN 1223-7027 NOTES ON A CONFORMAL CHARACTERIZATION OF 2-DIMENSIONAL LORENTZIAN MANIFOLDS WITH CONSTANT RICCI SCALAR CURVATURE 1 Nicol`o Cangiotti and Mattia Sensi2 We present a characterization of 2-dimensional Lorentzian manifolds with con-stant Ricci scalar curvature. It is well known that every 2-dimensional Lorentzian manifolds is conformally flat, so we rewrite the Ricci scalar curvature in terms of the conformal factor and we study the solutions of the corresponding differential equations. Several remarkable examples are provided. Keywords: Lorentzian manifolds, 2-dimensional manifolds, Minkowski space, conformal fac- tor, Ricci scalar curvature. MSC2020: 35L05, 53B30, 83C15. 1. Introduction Pseudo-Riemannian manifolds have historically been of interdisciplinary interest, and they have been extensively studied in many fields, from differential geometry to mathematical physics. In particular, there is a class which, for many reasons, has assumed an importance comparable to the one of Riemannian manifolds. It is indeed well known that, in the general theory of relativity, the spacetime has historically been modeled as a 4-dimensional Lorentzian manifold. Lorentzian manifolds represent a very interesting case study, which has been thoroughly investigated in mathematics [1, 2, 5] as much as in physics [9, 14, 18]. And, as it often happens in geometry, there are many properties which depend completely on the dimension of the manifold. This paper fits into the body of works that propose an analysis of the 2-dimensional Lorentzian geometry (see, for instance, [4, 6, 8, 13, 17, 19]). Our idea is the following: starting from the fact that every 2-dimensional Lorentzian manifold is conformally flat, we show a very simple way to construct Lorentzian manifolds with constant Ricci scalar curvature by deriving different expressions of the conformal factor. -
© Cambridge University Press Cambridge
Cambridge University Press 978-0-521-86069-7 - String Theory and M-Theory: A Modern Introduction Katrin Becker, Melanie Becker, and John H. Schwarz Index More information Index acceleration equation, 528 multiloop, 91 action off-shell, 105 p-brane, 7, 19, 23 one-loop, 94 sigma-model form, 29 Shapiro–Virasoro, 91 D0-brane, 149, 185 anomalous dimension, 653, 672 Einstein–Hilbert, 301 BMN operator, 680 in D dimensions, 550 anomaly Einstein–Maxwell, 560 characteristic class, 174, 176, 177 Nambu–Goto, 24, 26, 27, 155 conformal, 62, 66, 76, 658 point-particle, 18 form, 174, 178 nonrelativistic limit, 20 hexagon diagram, 170 Polyakov, 26 inflow mechanism, 183 string sigma-model, 26, 30, 37 M5-brane, 184 cosmological constant term, 28 NS5-brane, 184 supergravity in 11 dimensions, 304 R symmetry, 657, 673, 676 type I supergravity, 318 superconformal, 143 type IIA supergravity, 311 triangle diagram, 170 type IIB supergravity, 314, 317 anomaly cancellation, 9, 421 ADE E8 × E8,179 classification of singularities, 360, 436 SO(16) × SO(16), 292 Dynkin diagrams, 423 11-dimensional supergravity, 181 gauge group, 437 at M-theory boundary, 182 groups, 422, 423 heterotic M-theory, 470 subgroups of SU(2), 437 in six dimensions, 182, 426, 454 adjunction formula, 370 R symmetry, 672 Adler–Bardeen theorem, 170 type I superstring theory, 176 AdS/CFT duality, 15, 330, 612, 638 type IIB superstring theory, 175 for D3-branes, 638, 642 anthropic principle, 522 for half-BPS states, 654 anthropic reasoning, 15, 522 for M2-branes, 643 anti-de Sitter space, 15, 375, 377,