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Riemannian metrics on 2D-manifolds related to the Euler−Poinsot rigid body motion

Bernard Bonnard, Olivier Cots, Jean-Baptiste Pomet, Nataliya Shcherbakova (2014)

ESAIM: Control, Optimisation and Calculus of Variations

The Euler−Poinsot rigid body motion is a standard mechanical system and it is a model for left-invariant Riemannian metrics on SO(3). In this article using the Serret−Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover, the metric can be restricted to a 2D-surface, and the conjugate points of this metric are evaluated using recent works on surfaces of revolution. Another related 2D-metric on S2 associated to the...

Riemannian Polyhedra and Liouville-Type Theorems for Harmonic Maps

Zahra Sinaei (2014)

Analysis and Geometry in Metric Spaces

This paper is a study of harmonic maps fromRiemannian polyhedra to locally non-positively curved geodesic spaces in the sense of Alexandrov. We prove Liouville-type theorems for subharmonic functions and harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau Theorem on a complete pseudomanifolds with non-negative Ricci curvature. Then we study 2-parabolic admissible Riemannian polyhedra and prove some vanishing results on them.

Riemannian semisymmetric almost Kenmotsu manifolds and nullity distributions

Yaning Wang, Ximin Liu (2014)

Annales Polonici Mathematici

We consider an almost Kenmotsu manifold M 2 n + 1 with the characteristic vector field ξ belonging to the (k,μ)’-nullity distribution and h’ ≠ 0 and we prove that M 2 n + 1 is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant sectional curvature -4 and a flat n-dimensional manifold, provided that M 2 n + 1 is ξ-Riemannian-semisymmetric. Moreover, if M 2 n + 1 is a ξ-Riemannian-semisymmetric almost Kenmotsu manifold such that ξ belongs to the (k,μ)-nullity distribution, we prove that M 2 n + 1 is...

Riemannian symmetries in flag manifolds

Paola Piu, Elisabeth Remm (2012)

Archivum Mathematicum

Flag manifolds are in general not symmetric spaces. But they are provided with a structure of ℤ 2 k -symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. The conditions for a metric adapted to the ℤ 2 2 -symmetric structure to be naturally reductive are detailed for the flag manifold S O ( 5 ) / S O ( 2 ) × S O ( 2 ) × S O ( 1 ) .

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