arXiv subjects

Rafael Santamaría

Publications and source records attributed to Rafael Santamaría.

10 recordsLinked to original sources

arXiv2026

On the triviality of the generalized tangent bundle

We study the relations between the triviality of the tangent bundle $TM$ and the generalized tangent bundle $\mathbb{T}M = TM\oplus T^*M$ of a manifold. We show that the generalized tangent bundle of a paralellizable manifold is trivial. We also prove that the converse implication does not hold, by studying the cases of the Möbius strip, spheres and projective spaces. Finally, we relate the triviality of the generalized tangent bundle to generalized geometric structures.

math.DG↗

arXiv2025

About generalized complex structures on $\mathbb S^6$

We study the existence of generalized complex structures on the six-dimensional sphere $\mathbb S^6$. We work with the generalized tangent bundle $\mathbb T\mathbb S^6\to \mathbb S^6$ and define the integrability of generalized geometric structures in terms of the Dorfman bracket. Specifically, we prove that there is not a direct way to induce a generalized complex structure on $\mathbb S^6$ from its usual nearly Kähler structure inherited from the octonions product.

math.DG↗

arXiv2022

Metric polynomial structures on generalized geometry

In this document, we study the interaction between different geometric structures that can be defined as morphisms of sections of the generalized tangent bundle $\mathbb TM:= TM\oplus T^*M\to M$. In particular, we show the behaviour of various generalized polynomial structures with respect to different metrics on $\mathbb TM$, in the frame of $(α, \varepsilon)$-metric structures; and the commutation or anti-commutation of generalized polynomial structures, forming triple structures.

math.DG↗

arXiv2018

On nearly Kähler and Kähler-Codazzi type manifolds

Nearly Kähler and Kähler-Codazzi type manifolds are defined in a very similar way. We prove that nearly Kähler type manifolds have sense just in Hermitian and para-Hermitian contexts, and that Kähler-Codazzi type manifolds reduce to Kähler type manifolds in all the four Hermitian, para-Hermitian, Norden and product Riemannian geometries.

math.DG↗

arXiv2017

The canonical involution in the space of connections of a $(J^{2}=\pm 1)$-metric manifold

A $(J^{2}=\pm 1)$-metric manifold has an almost complex or almost product structure $J$ and a compatible metric $g$. We show that there exists a canonical involution in the set of connections on such a manifold, which allows to define a projection over the set of connections adapted to $J$. This projection sends the Levi Civita connection onto the first canonical connection. In the almost Hermitian case, it also sends the $\nabla^{-}$ connection onto the Chern connection, thus applying the line of metric connections defined by $\nabla ^{-}$ and the Levi Civita connections onto the line of canonical connections. Besides, it moves metric connections onto metric connections.

math.DG↗

arXiv2017

On the geometry of almost Golden Riemannian manifolds

An almost Golden Riemannian structure $(φ,g)$ on a manifold is given by a tensor field $φ$ of type (1,1) satisfying the Golden section relation $φ^{2}=φ+1$, and a pure Riemannian metric $g$, i.e., a metric satisfying $g(φX,Y)=g(X,φY)$. We study connections adapted to such a structure, finding two of them, the first canonical and the well adapted, which measure the integrability of $φ$ and the integrability of the $G$-structure corresponding to $(φ,g)$.

math.DG↗

arXiv2016

Distinguished connections on $(J^{2}=\pm 1)$-metric manifolds

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of $(J^2=\pm1)$-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted connections, named canonical connections, thus extending to almost Norden and almost product Riemannian manifolds the families introduced in almost Hermitian and almost para-Hermitian manifolds. We also prove that every connection studied in this paper is a canonical connection, when it exists and it is an adapted connection.

math.DG↗

arXiv2015

The well adapted connection of a $(J^{2}=\pm 1)$-metric manifold

In this paper, we study the well adapted connection attached to a $(J^{2}=\pm 1)$-metric manifold, proving it exists for any of the four geometries and obtaining a explicit formula as a derivation law. Besides we characterize the coincidence of the well adapted connection with the Levi Civita and the Chern connections.

math.DG↗

arXiv2006

Connections functorially attached to almost complex product structures

Manifolds endowed with three foliations pairwise transversal are known as 3-webs. Equivalently, they can be algebraically defined as biparacomplex or complex product manifolds, i.e., manifolds endowed with three tensor fields of type $(1,1)$, $F$, $P$ and $J=F \circ P$, where the two first are product and the third one is complex, and they mutually anti-commute. In this case, it is well known that there exists a unique torsion-free connection parallelizing the structure. In the present paper, we study connections attached to non-integrable almost biparacomplex manifolds.

math.DG↗

arXiv2004

The geometry of a bi-Lagrangian manifold

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyse the associated connections. Moreover, we introduce the problem of the intersection of two leaves, one of each foliation, through a point and show a lot of significative examples.

math.SG↗