Geometric methods for solving Codazzi and Monge-Amp�re equations

In the present paper it is considered a class V of 3-dimensional Riemannian manifolds M with a metric g and two affinor tensors q and S. It is defined another metric \bar{g} in M. The local coordinates of all these tensors are circulant matrices. It is found: 1)\ a relation between curvature tensors R and \bar{R} of g and \bar{g}, respectively; 2)\ an identity of the curvature tensor R of g in the case when the curvature tensor \bar{R} vanishes; 3)\ a relation between the sectional curvature of a 2-section of the type \{x, qx\} and the scalar curvature of M.

International Journal of Mathematics and Mathematical Sciences, 2002

We deal with a 2m-dimensional Riemannian manifold (M, g) structured by an affine connection and a vector field -, defining a --parallel connection. It is proved thatis both a torse forming vector field and an exterior concurrent vector field. Properties of the curvature 2-forms are established. It is shown that M is endowed with a conformal symplectic structure Ω anddefines a relative conformal transformation of Ω.

Monatshefte Fur Mathematik, 1990

For a given nondegenerate hypersurfaceM n in affine space ℝn+1 there exist an affine connection ∇, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=∇h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ∇ and a nondegenerate metrich on a differentiable manifoldM n such that ∇h is totally symmetric and satisfies the apolarity condition relative toh, canM n be locally immersed in ℝn+1 in such a way that (∇,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ∇. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (∇,h) is that the conjugate connection of ∇ relative toh is projectively flat.

Annals of Global Analysis and Geometry, 2015

A Riemann-Cartan manifold is a Riemannian manifold endowed with an affine connection which is compatible with the metric tensor. This affine connection is not necessarily torsion free. Under the assumption that the manifold is a homogeneous space, the notion of homogeneous Riemann-Cartan space is introduced in a natural way. For the case of the odd dimensional spheres S 2n+1 viewed as homogeneous spaces of the special unitary groups, the classical Nomizu's Theorem on invariant connections has permitted to obtain an algebraical description of all the connections which turn the spheres S 2n+1 into homogeneous Riemann-Cartan spaces. The expressions of such connections as covariant derivatives are given by means of several invariant tensors: the ones of the usual Sasakian structure of the sphere; an invariant 3-differential form coming from a 3-Sasakian structure on S 7 ; and the involved ones in the almost contact metric structure of S 5 provided by its natural embedding into the nearly Kähler manifold S 6. Furthermore, the invariant connections sharing geodesics with the Levi-Civita one have also been completely described. Finally, S 3 and S 7 are characterized as the unique odd-dimensional spheres which admit nontrivial invariant connections satisfying an Einstein-type equation.

European Journal of Pure and Applied Mathematics, 2011

In this study, we consider a manifold equipped with semi symmetric metric connection whose the torsion tensor satisfies a special condition. We investigate some properties of the Ricci tensor and the curvature tensor of this manifold. We obtain a necessary and sufficient condition for the mixed generalized quasi-constant curvature of this manifold. Finally, we prove that if the manifold mentioned above is conformally flat, then it is a mixed generalized quasi-Einstein manifold and we prove that if the sectional curvature of a Riemannian manifold with a semi symmetric metric connection whose the special torsion tensor is independent from orientation chosen, then this manifold is of a mixed generalized quasi constant curvature.

Israel Journal of Mathematics, 2009

We consider the pseudo-Euclidean space (R n , g), with n ≥ 3 and g ij = δ ij i , i = ±1, where at least one i = 1 and nondiagonal tensors of the form T = ij f ij dx i dx j such that, for i = j, f ij (x i , x j ) depends on x i and x j . We provide necessary and sufficient conditions for such a tensor to admit a metricḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R n , on the n-dimensional torus T n and on cylinders T k ×R n−k , that solve the Ricci equation or the Einstein equation.

Results in Mathematics, 2021

We continue our study of the mixed Einstein-Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metricaffine space and use them to derive Euler-Lagrange equations (which in the case of space-time are analogous to those in Einstein-Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler-Lagrange equations of the mixed Einstein-Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections.

Mathematics and Statistics, 2019

The Fundamental Theorem of Riemannian geometry states that on a Riemannian manifold there exist a unique symmetric connection compatible with the metric tensor. There are numerous examples of connections that even locally do not admit any compatible met-rics. A very important class of symmetric connections in the tangent bundle of a certain manifolds (afinnely flat) are the ones for which the curvature tensor vanishes. Those connections are locally metric. S.S. Chern conjectured that the Euler characteristic of an affinely flat manifold is zero. A possible proof of this long outstanding conjecture of S.S. Chern would be by verifying that the space of locally metric connections is path connected. In order to do so one needs to have practical criteria for the metrizability of a connection. In this paper we give necessary and sufficient conditions for a connection in a plane bundle above a surface to be locally metric. These conditions are easy to be verified using any local frame. Also, as a global result we give a necessary condition for two connections to be metric equivalent in terms of their Euler class.

Universal Journal of Mathematics and Applications, 2020

The aim of the present paper is to study the properties of Riemannian manifolds equipped with a projective semi-symmetric connection.

2020

We study the properties of Ricci curvature of ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the ${\mathfrak{g}}$-manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian ${\mathfrak{g}}$-manifolds.

The Annals of Mathematics, 2002

Conformal geometry in two dimensions is distinguished by its relationship to complex analysis. In higher dimensions the landscape becomes more com-plicated, and in the absence of some special structure (eg, Kiihler) even an extensive knowledge of the theory of ...

Abstract and Applied Analysis, 2013

The object of the present paper is to study -manifolds with vanishing quasi-conformal curvature tensor. -manifolds satisfying Ricci-symmetric condition are also characterized.

Facta Universitatis, Series: Mathematics and Informatics, 2021

The differential geometry of the tangent bundle is an effective domain of differential geometry which reveals many new problems in the study of modern differential geometry. The generalization of connection on any manifold to its tangent bundle is an application of differential geometry. Recently a new type of semi-symmetric non-metric connection on a Riemannian manifold has been studied and a relationship between Levi-Civita connection and semi-symmetric non-metric connection has been established. The various properties of a Riemannian manifold with relation to such connection have also been discussed. The present paper aims to study the tangent bundle of a new type of semi-symmetric non-metric connection on a Riemannian manifold. The necessary and sufficient conditions for projectively invariant curvature tensors corresponding to such connection are proved and show many basic results on the Riemannian manifold in the tangent bundle. Furthermore, the properties of group manifolds of the Riemannian manifolds with respect to the semi-symmetric non-metric connection in the tangent bundle have been studied. Moreover, theorems on the symmetry property of Ricci tensor and Ricci soliton in the tangent bundle are established.

Journal of Functional Analysis, 2006

The main result of the paper is a computation of the Ricci curvature of Diff(S 1 )/S 1 . Unlike earlier results on the subject, we do not use the Kähler structure symmetries to compute the Ricci curvature, but rather rely on classical finite-dimensional results of Nomizu et al. on Riemannian geometry of homogeneous spaces.