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.

A Note on Hyperconvexity in Riemannian Manifolds

1959 ◽  
Vol 11 ◽  
pp. 576-582
Author(s):  
Albert Nijenhuis
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Shortest Path ◽  
Riemannian Manifolds ◽  
Convex Set ◽  
Geodesic Segment ◽  
Positive Definite ◽  
Classical Theorem ◽  
Arc Length ◽  
Class C

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.

Curvature of K-contact Semi-Riemannian Manifolds

2014 ◽  
Vol 57 (2) ◽  
pp. 401-412 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Lorentzian Manifold ◽  
Constant Sectional Curvature ◽  
Conformally Flat ◽  
Reeb Vector ◽  
Reeb Vector Field

Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

scholarly journals Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations

1988 ◽  
Vol 8 (2) ◽  
pp. 215-239 ◽  
Author(s):  
Masahiko Kanai
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Closed Riemannian Manifold

AbstractWe are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.

scholarly journals Indefinite Almost Paracontact Metric Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Mukut Mani Tripathi ◽  
Erol Kılıç ◽  
Selcen Yüksel Perktaş ◽  
Sadık Keleş
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Sasakian Manifold ◽  
Constant Sectional Curvature ◽  
Sasakian Manifolds ◽  
Sasakian Structure

We introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an (ε)-para Sasakian structure. We show that, for an (ε)-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) (ε)-para Sasakian manifoldMnis locally isometric to a pseudohyperbolic spaceHνn(1)(resp., pseudosphereSνn(1)). At last, it is proved that for an (ε)-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.

A class of non-holonomic projective connections on sub-Riemannian manifolds

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1295-1303
Author(s):  
Yanling Han ◽  
Fengyun Fu ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Projective Transformation ◽  
Projective Connection

The authors define a semi-symmetric non-holonomic (SSNH)-projective connection on sub-Riemannian manifolds and find an invariant of the SSNH-projective transformation. The authors further derive that a sub-Riemannian manifold is of projective flat if and only if the Schouten curvature tensor of a special SSNH-connection is zero.

scholarly journals On curvature tensors of Norden and metallic pseudo-Riemannian manifolds

2019 ◽  
Vol 6 (1) ◽  
pp. 150-159 ◽  
Author(s):  
Adara M. Blaga ◽  
Antonella Nannicini
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Riemannian Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Metallic Structures ◽  
Bisectional Curvature ◽  
Curvature Tensors

AbstractWe study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.

THE HOMOGENEOUS LIFT TO THE (1, 1)-TENSOR BUNDLE OF A RIEMANNIAN METRIC

2013 ◽  
Vol 10 (04) ◽  
pp. 1350006 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
HASSAN NASRABADI ◽  
AKBAR TAYEBI
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Differentiable Manifold ◽  
Riemannian Metric ◽  
Riemannian Curvature ◽  
Riemannian Curvature Tensor ◽  
Geometric Objects ◽  
Levi Civita Connection ◽  
Tensor Bundle

By using a Riemannian metric on a differentiable manifold, the homogeneous lift metric is introduced on the (1, 1)-tensor bundle of the Riemannian manifold. Some geometric objects related to this metric, such as the Levi-Civita connection, Riemannian curvature tensor and sectional curvature are calculated. Also, a para-Nordenian structure on the (1, 1)-tensor bundle with this metric is constructed and interesting properties of this structure are studied.

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.

Prescribed k-Curvature Problems on Complete Noncompact Riemannian Manifolds

2018 ◽  
Vol 2020 (23) ◽  
pp. 9559-9592
Author(s):  
Jixiang Fu ◽  
Weimin Sheng ◽  
Lixia Yuan
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Elliptic Partial Differential Equations ◽  
Nonlinear Elliptic ◽  
Noncompact Riemannian Manifolds ◽  
Curvature Tensors ◽  
Curvature Problem ◽  
Negative Sectional Curvature

Abstract To study the prescribed $k$-curvature problem of 2nd-order symmetric curvature tensors on complete noncompact Riemannian manifolds, we consider a class of fully nonlinear elliptic partial differential equations. It is proved that on a Riemannian manifold with negative sectional curvature and Ricci curvature bounded from below, the equation is solvable provided that all the eigenvalues of the tensor are negative. The result is applicable to the prescribed $k$-curvature problems of modified Schouten tensor and Bakry–Émery Ricci tensor.

scholarly journals Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II

2020 ◽  
Vol 28 (1) ◽  
pp. 165-179
Author(s):  
Luca Sabatini
Keyword(s):  
Riemannian Manifold ◽  
Lower Bound ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Injectivity Radius ◽  
Topological Invariants ◽  
Manifolds With Boundary ◽  
Laplacian Spectrum ◽  
Convex Boundary

AbstractWe present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

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