The Gauss-Bonnet Integrand for a Class of Riemannian Manifolds Admitting Two Orthogonal Complementary Foliations

1983 ◽  
Vol 26 (3) ◽  
pp. 358-364 ◽  
Author(s):  
O. Gil-Medrano ◽  
A. M. Naveira
Keyword(s):  
Sectional Curvature ◽  
Euler Characteristic ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Tensor Field ◽  
General Assumption ◽  
Totally Geodesic ◽  
Geodesic Foliation ◽  
Characteristic Connection ◽  
Totally Geodesic Foliation

AbstractWith the general assumption that the manifold admits two orthogonal complementary foliations, one of which is totally geodesic, we study the components of the curvature tensor field of the characteristic connection.In the case where the manifold is compact, orientable of dimension 6 or 8 and the dimension of the totally geodesic foliation is 4, we relate the sign of the Euler characteristic of the manifold and that of the sectional curvature of the leaves of both foliations.

scholarly journals On the number of diffeomorphism classes in a certain class of Riemannian manifolds

1985 ◽  
Vol 97 ◽  
pp. 173-192 ◽  
Author(s):  
Takao Yamaguchi
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Homotopy Types

The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.

scholarly journals Metric f-Contact Manifolds Satisfying the (κ, μ)-Nullity Condition

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 891
Author(s):  
Alfonso Carriazo ◽  
Luis M. Fernández ◽  
Eugenia Loiudice
Keyword(s):  
Explicit Expression ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Tensor Field ◽  
Contact Manifold ◽  
Contact Manifolds

We prove that if the f-sectional curvature at any point of a ( 2 n + s ) -dimensional metric f-contact manifold satisfying the ( κ , μ ) nullity condition with n > 1 is independent of the f-section at the point, then it is constant on the manifold. Moreover, we also prove that a non-normal metric f-contact manifold satisfying the ( κ , μ ) nullity condition is of constant f-sectional curvature if and only if μ = κ + 1 and we give an explicit expression for the curvature tensor field in such a case. Finally, we present some examples.

An investigation on the existence of warped product irrotational screen-real lightlike submanifolds of metallic semi-Riemannian manifolds

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Gauree Shanker ◽  
Ankit Yadav
Keyword(s):  
Riemannian Manifolds ◽  
Design Methodology ◽  
Warped Product ◽  
Sufficient Conditions ◽  
Totally Geodesic ◽  
Content Type ◽  
Lightlike Submanifolds ◽  
Totally Geodesic Foliation ◽  
Is Design ◽  
Lightlike Submanifold

PurposeThe purpose of this paper is to study the geometry of screen real lightlike submanifolds of metallic semi-Riemannian manifolds. Also, the authors investigate whether these submanifolds are warped product lightlike submanifolds or not.Design/methodology/approachThe paper is design as follows: In Section 3, the authors introduce screen-real lightlike submanifold of metallic semi Riemannian manifold. In Section 4, the sufficient conditions for the radical and screen distribution of screen-real lightlike submanifolds, to be integrable and to be have totally geodesic foliation, have been established. Furthermore, the authors investigate whether these submanifolds can be written in the form of warped product lightlike submanifolds or not.FindingsThe geometry of the screen-real lightlike submanifolds has been studied. Also various results have been established. It has been proved that there does not exist any class of irrotational screen-real r-lightlike submanifold such that it can be written in the form of warped product lightlike submanifolds.Originality/valueAll results are novel and contribute to further study on lightlike submanifolds of metallic semi-Riemannian manifolds.

scholarly journals The weighted mixed curvature of a foliated manifold

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1097-1105
Author(s):  
Vladimir Rovenski
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Foliation ◽  
Foliated Manifold ◽  
Totally Geodesic ◽  
Geodesic Foliation ◽  
New Concepts ◽  
Sectional And Ricci Curvatures ◽  
Totally Geodesic Foliation ◽  
Foliated Riemannian Manifold

We introduce the weighted mixed curvature of an almost product (e.g. foliated) Riemannian manifold equipped with a vector field. We define several qth Ricci type curvatures, which interpolate between the weighed sectional and Ricci curvatures. New concepts of the ?mixed-curvature-dimension condition? and ?synthetic dimension of a distribution? allow us to renew the estimate of the diameter of a compact Riemannian foliation and splitting results for almost product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. We also study the Toponogov?s type conjecture on dimension of a totally geodesic foliation with positive weighted mixed sectional curvature.

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