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 Hamiltonian constraint equation becomes a generalisation of the σk-Yamabe problem, i.e., the prescription of a linear combination of the k-curvatures of the manifold.
Abstract: This paper is devoted to the study of the constraint equations of the Lovelock gravity theories. In the case of a conformally flat, time-symmetric, and space-like manifold, we show that the Hamiltonian constraint equation becomes a generalisation of the σk-Yamabe problem. That is to say, the prescription of a linear combination of the σk-curvatures of the manifold. We search solutions in a conformal class for a compact manifold. Using the existing results on the σk-Yamabe problem, we describe some cases in which they can be extended to this new problem. This requires to study the concavity of some polynomial. We do it in two ways: regarding the concavity of a root of this polynomial, which is connected to algebraic properties of the polynomial; and seeking analytically a concavifying function. This gives several cases in which a conformal solution exists. Finally we show an implicit function theorem in the case of a manifold with negative scalar curvature, and find a conformal solution when the Lovelock theories are close to General Relativity.
4 citations
TL;DR: In this paper, it was shown that if a vector field V on a contact metric manifold of dimension 2n+1 leaves the tensor field invariant, then V is an infinitesimal harmonic transformation.
Abstract: We prove that if a vector field V on a contact metric manifold \({M(\varphi, \xi, \eta, g)}\) of dimension (2n+1) leaves the tensor field \({\varphi}\) invariant, then V is an infinitesimal harmonic transformation. Next, we study contact metric manifolds admitting a vector field V that leaves the structure tensor \({\varphi}\) invariant and satisfies different conditions, namely (1) \({Q\varphi = \varphi Q}\), (2) M is Jacobi \({(k, \mu)}\)-contact manifold, (3) \({R(X, Y)\xi = 0}\), for any vector fields X, Y orthogonal to \({\xi}\) and (4) \({\pounds_{V}C = 0}\), where C is the conformal curvature tensor.
4 citations
TL;DR: In this article, the authors characterize paracontact metric manifolds conceding almost Yamabe solitons and establish a few fascinating results of such soliton. But these results are restricted to N(k, ε)-parAContact manifolds.
Abstract: The goal of the current paper is to characterize paracontact metric manifolds conceding $$\delta $$
-almost Yamabe solitons. A few fascinating results of such solitons are established. Specifically, we classify $$\delta $$
-almost Yamabe solitons on $$(k,\mu )$$
and N(k)-paracontact metric manifolds.
4 citations
TL;DR: In this article, the maximal holomorphic tangent subspace is (n − 1)-dimensional and a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces is given.
Abstract: We treat n-dimensional compact minimal submanifolds of complex projective space when the maximal holomorphic tangent subspace is (n − 1)-dimensional and we give a sufficient condition for such submanifolds to be tubes over totally geodesic complex subspaces.
3 citations