scholarly journals Horizontal diameter of unit spheres with polar foliations and infinitesimally polar actions

2021 ◽  
Vol 32 (04) ◽  
pp. 2150018
Author(s):  
Yi Shi
Keyword(s):  
Riemannian Manifold ◽  
Unit Sphere ◽  
Riemannian Foliation ◽  
Horizontal Curve ◽  
Polar Action ◽  
Polar Actions ◽  
Horizontal Diameter ◽  
Singular Riemannian Foliation ◽  
Polar Foliations

For a singular Riemannian foliation [Formula: see text] on a Riemannian manifold, a curve is called horizontal if it meets the leaves of [Formula: see text] perpendicularly. For a singular Riemannian foliation [Formula: see text] on a unit sphere [Formula: see text], we show that if [Formula: see text] is a polar foliation or if [Formula: see text] is given by the orbits of an infinitesimally polar action, then the horizontal diameter of [Formula: see text] is [Formula: see text], i.e. any two points in [Formula: see text] can be connected by a horizontal curve of length [Formula: see text].

scholarly journals Examples and classification of Riemannian submersions satisfying a basic equality

2005 ◽  
Vol 72 (3) ◽  
pp. 391-402 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Unit Sphere ◽  
Isometric Immersion ◽  
Riemannian Submersion ◽  
Sharp Inequality ◽  
Riemannian Submersions ◽  
Equality Case ◽  
Totally Geodesic ◽  
Basic Equality

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

On the horizontal diameter of the unit sphere

2017 ◽  
Vol 110 (1) ◽  
pp. 91-97
Author(s):  
Yi Shi ◽  
Zhiqi Xie
Keyword(s):  
Unit Sphere ◽  
Horizontal Diameter

POLAR ACTIONS ON BERGER SPHERES

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1019-1023 ◽  
Author(s):  
ANTONIO J. DI SCALA
Keyword(s):  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Type Theorem ◽  
Torus Action ◽  
Polar Action ◽  
Polar Actions ◽  
Berger Spheres ◽  
Berger Sphere ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive

The object of this article is to study a torus action on a so-called Berger sphere. We also make some comments on polar actions on naturally reductive homogeneous spaces. Finally, we prove a rigidity-type theorem for Riemannian manifolds carrying a polar action with a fix point.

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.

scholarly journals EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN

2010 ◽  
Vol 53 (2) ◽  
pp. 321-332 ◽  
Author(s):  
SUN HEJUN ◽  
QI XUERONG
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Eigenvalue Problem ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Complete Riemannian Manifold ◽  
Quadratic Polynomial ◽  
Polynomial Operator ◽  
Eigenvalue Estimates ◽  
The Mean

AbstractFor a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

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