Related papers: Jacobi fields on normal homogeneous Riemannian man…

Riemannian geodesic orbit spaces (G/H,g) are natural generalizations of symmetric spaces, defined by the property that their geodesics are orbits of one-parameter subgroups of G. We study the geodesic orbit spaces of the form (G/S,g), where…

In this paper we have obtained evolution of some geometric quantities on a compact Riemannian manifold $M^n$ when the metric is a Yamabe soliton. Using these quantities we have obtained bound on the soliton constant. We have proved that the…

We investigate geometric properties of homogeneous parabolic geometries with generalized symmetries. We show that they can be reduced to a simpler geometric structures and interpret them explicitly. For specific types of parabolic…

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally…

The existence of a Killing symmetry in a gauge theory is equivalent to the addition of extra Hamiltonian constraints in its phase space formulation, which imply restrictions both on the Dirac observables (the gauge invariant physical…

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces…

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected…

We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.

We prove that equivariant, holomorphic embeddings of Hermitian symmetric spaces are totally geodesic (when the image is not of exceptional type).

A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces…

This paper provides a characterization of homogeneous curves on a geometric flag manifold which are geodesic with respect to any invariant metric. We call such curves homogeneous equigeodesics. We also characterize homogeneous equigeodesics…

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on…

The harmonic action functional allows a natural generalisation to semi-Riemannian supergeometry, referred to as superharmonic action, which resembles the supersymmetric sigma models studied in high energy physics. We show that Killing…

In this paper, we define Jacobi fields for nonholonomic mechanics using a similar characterization than in Riemannian geometry. We give explicit conditions to find Jacobi fields and finally we find the nonholonomic Jacobi equations in two…

It is well known since Jacobi that the geodesic flow of the ellipsoid is "completely integrable", which means that the geodesic orbits are described in a certain explicit way. However, it does not directly indicate that any global behavior…

We consider a (pseudo)Riemannian manifold of arbitrary dimension. The Hamilton-Jacobi equation for geodesic Hamiltonian admits complete separation of variables for some (separable) metrics in some (separable) coordinate systems. Separable…

We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped…

A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for…

In a previous paper, we showed that any Jacobi field along a harmonic map from the 2-sphere to the complex projective plane is integrable (i.e., is tangent to a smooth variation through harmonic maps). In this paper, in contrast, we show…

We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that…