Jose Carmelo Gonzalez-Davila

Universidad de La Laguna | ULL · Department of Fundamental Mathematics

A positive answer is given to the existence of Sasakian structures on the tangent sphere bundle of some Riemannian manifold whose sectional curvature is not constant. Among other results, it is proved that the tangent sphere bundle $$T_{r}(G/K),$$ T r ( G / K ) , for any $$r> 0,$$ r > 0 , of a compact rank-one symmetric space G / K , not necessaril...

A positive answer is given to the existence of Sasakian structures on the tangent sphere bundle of some Riemannian manifold whose sectional curvature is not constant. Among other results, it is proved that the tangent sphere bundle Tr(G/K), for any r > 0, of a compact rank-one symmetric space G/K, not necessarily of constant sectional curvature, ad...

We give an explicit description of all complete $G$-invariant Ricci-flat K\"ahler metrics on the tangent bundle $T(G/K)\cong G^\bbC/K^\bbC$ of rank-one Riemannian symmetric spaces $G/K$ of compact type, in terms of associated vector-functions.

We give a description of all $G$-invariant Ricci-flat K\"ahler metrics on the canonical complexification of any compact Riemannian symmetric space $G/K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G/K)$. As the simplest application, we obtain the explicit description of the set of all complete $\mathrm{SO}(3)$-invaria...

We construct special classes of totally geodesic almost regular foliations, namely, complex radial foliations in Hermitian manifolds and quaternionic radial foliations in quaternionic Kähler manifolds, and we give criteria for their harmonicity and minimality. Then examples of these foliations on complex and quaternionic space forms, which are harm...

We consider the energy of smooth generalized distributions and also of singular foliations on compact Riemannian manifolds for which the set of their singularities consists of a finite number of isolated points and of pairwise disjoint closed submanifolds. We derive a lower bound for the energy of all $q$-dimensional almost regular distributions, f...

We investigate the existence of non-symmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G/K$ and reductive decomposition $\mathfrak{g}...

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and we obtain the classification o...

Lie groups with a cyclic left-invariant metric are Lie groups which are in some way far from being bi-invariant Lie groups, in a sense made explicit in terms of Tricerri and Vanhecke's homogeneous structures. The semi-simple and solvable cases are studied. The classification of connected and simply-connected Lie groups with a cyclic left-invariant...

Cyclic metric Lie groups are Lie groups equipped with a left-invariant metric which is in some way far from being biinvariant, in a sense made explicit in terms of Tricerri and Vanhecke’s homogeneous structures. The semisimple and solvable cases are studied. We extend to the general case, Kowalski–Tricerri’s and Bieszk’s classifications of connecte...

We consider a q-dimensional distribution as a section of the Grassmannian bundle G(q)(M-n) of q-planes and we derive, in terms of the intrinsic torsion of the corresponding S(O(q) x O(n-q))-structure, the conditions that this map must satisfy in order to be critical for the functionals energy and volume. Using this it is shown that invariant Rieman...

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 rank one normal homogeneous space with the property that the isotropy action is variational complete is a rank on...

We proceed further in the study of harmonicity for almost contact metric structures already initiated by Vergara-Díaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric structures. Additionally, w...

The structure of nearly K\"ahler manifolds was studied by Gray in several papers. More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a strict and complete nearly K\"ahler manifold is locally a Riemannian product of homogeneous nearly K\"ahler spaces, twistor spaces over quaternionic K{\"ahle...

We prove that a compact Riemannian 3-symmetric space is globally symmetric if every Jacobi field along a geodesic vanishing at two points is the restriction to that geodesic of a Killing field induced by the isotropy action or, in particular, if the isotropy action is variationally complete.

We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This is a positive answer for a rev...

The geometry of Riemannian symmetric spaces is really richer than that of Riemannian homogeneous spaces. Nevertheless, there exists a large literature of special classes of homogeneous Riemannian manifolds with an important list of features which are typical for a Riemannian symmetric space. Normal homogeneous spaces, naturally reductive homogeneou...

We study the Jacobi osculating rank of geodesics on naturally reductive homogeneous manifolds and we apply this theory to the 3-dimensional case. Here, each non-symmetric, simply connected naturally reductive 3-manifold can be given as a principal bundle over a surface of constant curvature, such that the curvature of its horizontal distribution is...

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, <,>) equipped with the Sasaki metric and we discuss the characterising condition for critical points. Likewise, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate f...

For closed and connected subgroups G of SO(n), we study the energy functional on the space of G-structures of a (compact) Riemannian manifold M, where G-structures are considered as sections of the quotient bundle O(M)/G. Then, we deduce the corresponding first and second variation formulae and the characterising conditions for critical points by m...

Without Abstract

We derive existence results of isotropic Jacobi fields on naturally reductive spaces and we prove that a naturally reductive space (M,g) of dimension ≤5 with the property that all Jacobi fields vanishing at two points are Tr(M,∇ ˜)-isotropic, for some adapted canonical connection ∇ ˜ and where Tr(M,∇ ˜) denotes the corresponding transvection group,...

We consider an oriented distribution as a section of the corresponding Grassmann bundle and, by computing the tension of this map for conveniently chosen metrics, we obtain the conditions which the distribution must satisfy in order to be critical for the functionals related to the volume or the energy of the map. We show that the three-dimensional...

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds and...

We study the stability of the geodesic flow ξ as a critical point for the energy functional when the base space is a compact orientable quotient of a two-point homogeneous space.

We provide a new characterization of invariant harmonic unit vector fields on Lie groups endowed with a left-invariant metric. We use it to derive existence results and to construct new examples on Lie groups equipped with a bi-invariant metric, on three-dimensional Lie groups, on generalized Heisenberg groups, on Damek-Ricci spaces and on particul...

We study the stability and instability of harmonic and minimal unit vector fields and the existence of absolute minima for the energy and volume functional on three-dimensional compact manifolds, in particular on compact quotients of unimodular Lie groups.

We study the stability of the geodesic flow ξ as a critical point for the en-ergy functional when the base space is a compact orientable quotient of a two-point homogeneous space.

We consider three-dimensional contact metric manifolds whose unit characteristic vector field is harmonic or minimal. A unit vector field ξ on a Riemannian manifold (M, g) determines a map from (M, g) to its unit tangent bundle (T1M, gS) equipped with the Sasaki metric gS. It also determines a submanifold of this unit tangent sphere bundle. When M...

We consider unit vector fields on homogeneous Riemannian manifolds (M = G/Go, g) which are G-invariant. We derive a criterion for the minimality and for the harmonicity of such vector fields by means of the infinitesimal models which correspond to (locally) homogeneous spaces and which are determined by using homogeneous structures. This leads to t...

We provide a series of examples of Riemannian manifoldsequipped with a minimal unit vector field.

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex s...

We prove that any simply connected Riemannian manifold which is equipped with a complete unit Killing vector field such that the reflections with respect to the flow lines of that field can be extended to global isometries, is a weakly symmetric space.

We determine explicitly Jacobi vector fields on complete locally Killing transversally symmetric spaces and normal flow space forms. We use the Jacobi operator, the Jacobi vector fields and the shape operator of geodesic spheres to derive several characterizations of these spaces.

We consider the shape operator of tubular hypersurfaces about geodesies in Riemannian manifolds which are equipped with a unit Killing vector field. We derive some characteristic properties for the special subclass of normal flow space forms.

It is still an open problem whether Riemannian manifolds all of whose local geodesic symmetries are volume-preserving (that is, D'Atri spaces) and Riemannian manifolds such that the Jacobi operators has constant eigenvalues along the corresponding geodesics (that is, C-spaces) are locally homogeneous. In this paper, we classify these spaces in dime...

We provide some new examples of weakly symmetric spaces inside the class of complete, simply connected Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to its flow lines are global isometries.

We begin a study of invariant isometric immersions into Riemannian manifolds (M, g) equipped with a Riemannian flow generated by a unit Killing vector field ξ. We focus our attention on those (M, g) where ξ is complete and such that the reflections with respect to the flow lines are global isometries (that is, (M, g) is a Killing-transversally symm...

We treat Killing-transversally symmetric spaces (briclly. KTS-spaces), that is Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to the flow lines of that field can be extended to global isometries. Sucha manifolds are homogencous spaces equipped with a naturally reductive homogeneous str...

We extend the Gelfand-like characterization of simply connected φ-symmetric spaces to the broader class of simply connected contact Killing-transversally symmetric spaces.

It is obtained a complete classification for almost contact metric manifolds through the study of the covariant derivative of the fundamental 2- form on those manifolds.

All the homogeneous structures on the generalized Heisenberg group H(p, 1) are found, obtaining a one-parameter family of quasi-Sasakian homogeneous structures on this group.

A smooth unit vector field V on a Riemannian manifold (M,g) determines a map from (M,g) into its unit tangent bundle (T 1 ,M,g S ) equipped with the Sasaki metric g S and its image is a submanifold of it. When M is closed and orientable, this gives rise to the consideration of two functionals on the set of smooth unit vector fields: the volume of t...