scholarly journals The Mixed Scalar Curvature of a Twisted Product Riemannian Manifolds and Projective Submersions

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 527
Author(s):  
Vladimir Rovenski ◽  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Warped Products ◽  
Twisted Product

In the present paper, we study twisted and warped products of Riemannian manifolds. As an application, we consider projective submersions of Riemannian manifolds, since any Riemannian manifold admitting a projective submersion is necessarily a twisted product of some two Riemannian manifolds.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

2019 ◽  
Vol 19 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Sergey Stepanov ◽  
Irina Tsyganok
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Type Theorem ◽  
Complete Riemannian Manifold ◽  
Warped Products ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Totally Umbilical

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

scholarly journals The geometry of warped product singularities

2017 ◽  
Vol 14 (02) ◽  
pp. 1750024 ◽  
Author(s):  
Ovidiu Cristinel Stoica
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Big Bang ◽  
Warping Function ◽  
Warped Products ◽  
Riemann Curvature ◽  
The Big Bang

In this article, the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Applications include cosmological models and black holes solutions with semi-regular singularities. Such singularities are compatible with a certain reformulation of the Einstein equation, which in addition holds at semi-regular singularities too.

scholarly journals Volume and macroscopic scalar curvature

2021 ◽  
Author(s):  
Sabine Braun ◽  
Roman Sauer
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Betti Numbers ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Simplicial Volume ◽  
Positive Scalar ◽  
Curvature Bound

AbstractWe prove the macroscopic cousins of three conjectures: (1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, (2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, (3) a conjectural bound of $$\ell ^2$$ ℓ 2 -Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of 1-balls in the universal cover.

An Integral Formula for a Riemannian Manifold with $k>2$ Singular Distributions

2021 ◽  
Vol 22 ◽  
pp. 253-262
Author(s):  
Vladimir Rovenski
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Integral Formula ◽  
Warped Products ◽  
Riemannian Foliations ◽  
Singular Riemannian Foliations

Mathematicians have shown interest in manifolds endowed with several distributions, e.g., webs composed of different regular foliations and multiply warped products, as well as distributions having variable dimensions (e.g., singular Riemannian foliations). In this paper, we extend our previous study of the mixed scalar curvature of two orthogonal singular distributions for the case of $k>2$ singular (or regular) pairwise orthogonal distributions, prove an integral formula with this kind of curvature, and illustrate it by characterizing autoparallel singular distributions.

The Local Möbius Equation and Decomposition Theorems in Riemannian Geometry

2002 ◽  
Vol 45 (3) ◽  
pp. 378-387
Author(s):  
Manuel Fernández-López ◽  
Eduardo García-Río ◽  
Demir N. Kupeli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Warped Product ◽  
Product Structure ◽  
Twisted Product ◽  
Partial Differential ◽  
Decomposition Theorems

AbstractA partial differential equation, the local Möbius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local Möbius equation and an additional partial differential equation.

scholarly journals On a Metric Affine Manifold with Several Orthogonal Complementary Distributions

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 229
Author(s):  
Vladimir Rovenski ◽  
Sergey E. Stepanov
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Scalar Curvature ◽  
Existence Theorems ◽  
Linear Connection ◽  
Product Structure ◽  
Warped Products ◽  
Affine Manifold ◽  
Integral Formulas ◽  
Levi Civita Connection

A Riemannian manifold endowed with k>2 orthogonal complementary distributions (called here an almost multi-product structure) appears in such topics as multiply twisted or warped products and the webs or nets composed of orthogonal foliations. In this article, we define the mixed scalar curvature of an almost multi-product structure endowed with a linear connection, and represent this kind of curvature using fundamental tensors of distributions and the divergence of a geometrically interesting vector field. Using this formula, we prove decomposition and non-existence theorems and integral formulas that generalize results (for k=2) on almost product manifolds with the Levi-Civita connection. Some of our results are illustrated by examples with statistical and semi-symmetric connections.

Isometry of Riemannian Manifolds to Spheres, II

1976 ◽  
Vol 28 (1) ◽  
pp. 63-72 ◽  
Author(s):  
Neill H. Ackerman ◽  
C. C. Hsiung
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Symmetric Matrix ◽  
Riemann Tensor ◽  
Positive Definite ◽  
Inverse Matrix ◽  
Covariant Differentiation

Let Mn be a Riemannian manifold of dimension n ≧ 2 and class C3, (gtj) the symmetric matrix of the positive definite metric of Mn, and (gij) the inverse matrix of (gtj), and denote by and R = gijRij the operator of covariant differentiation with respect to gij, the Riemann tensor, the Ricci tensor and the scalar curvature of Mn respectively.

scholarly journals The scalar curvature of the tangent bundle of a Finsler manifold

2011 ◽  
Vol 89 (103) ◽  
pp. 57-68
Author(s):  
Aurel Bejancu ◽  
Reda Farran
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Degree Zero ◽  
Necessary And Sufficient

Let Fm = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on the slit tangent bundle TM0 = TM \{0} of M. We express the scalar curvature ?~ of the Riemannian manifold (TM0,G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ?~ to be a positively homogenenous function of degree zero with respect to the fiber coordinates of TM0. Finally, we obtain characterizations of Landsberg manifolds, Berwald manifolds and Riemannian manifolds whose ?~ satisfies the above condition.

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

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