Nonexistence of Nontrivial □ � � -Harmonic 1-Forms on a Complete Foliated Riemannian Manifold

1980 ◽  
Vol 262 (2) ◽  
pp. 429 ◽  
Author(s):  
Haruo Kitahara
Keyword(s):  
Riemannian Manifold ◽  
Foliated Riemannian Manifold

Green's theorem on a foliated Riemannian manifold and its applications

1990 ◽  
Vol 56 (3-4) ◽  
pp. 239-245 ◽  
Author(s):  
S. Yorozu ◽  
T. Tanemura
Keyword(s):  
Riemannian Manifold ◽  
Green's Theorem ◽  
Green’S Theorem ◽  
Foliated Riemannian Manifold

scholarly journals Dynamics of the geodesic flow of a foliation

1988 ◽  
Vol 8 (4) ◽  
pp. 637-650 ◽  
Author(s):  
Paweł G. Walczak
Keyword(s):  
Riemannian Manifold ◽  
Lyapunov Exponents ◽  
Geodesic Flow ◽  
Smooth Measure ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Foliated Riemannian Manifold

AbstractThe geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.

scholarly journals Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold

1997 ◽  
Vol 62 (1) ◽  
pp. 46-63 ◽  
Author(s):  
Grant Cairns ◽  
Richard H. Escobales
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Final Section ◽  
General Context ◽  
Topological Properties ◽  
Normal Plane ◽  
Plane Field ◽  
The Mean ◽  
Foliated Riemannian Manifold

AbstractFor foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

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.

Improved algorithm of preserving global and local properties based on Riemannian manifold learning

2011 ◽  
Vol 30 (12) ◽  
pp. 3301-3303
Author(s):  
Wei WANG ◽  
Du-yan BI ◽  
Lei XIONG
Keyword(s):  
Riemannian Manifold ◽  
Manifold Learning ◽  
Local Properties ◽  
Global And Local ◽  
Improved Algorithm

Golden Riemannian structures on the tangent bundle with g-natural metrics

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2543-2554
Author(s):  
E. Peyghan ◽  
F. Firuzi ◽  
U.C. De
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Riemannian Metric

Starting from the g-natural Riemannian metric G on the tangent bundle TM of a Riemannian manifold (M,g), we construct a family of the Golden Riemannian structures ? on the tangent bundle (TM,G). Then we investigate the integrability of such Golden Riemannian structures on the tangent bundle TM and show that there is a direct correlation between the locally decomposable property of (TM,?,G) and the locally flatness of manifold (M,g).

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