arXiv subjects

Antonio J. Di Scala

Publications and source records attributed to Antonio J. Di Scala.

46 records · Page 3Linked to original sources

arXiv2008

Connected subgroups of SO(2,n) acting irreducibly on $\R^{2,n}$

We classify all connected subgroups of SO(2,n) that act irreducibly on $\R^{2,n}$. Apart from $SO_0(2,n)$ itself these are $U(1,n/2)$, $SU(1,n/2)$, if $n$ even, $S^1\cdot SO(1,n/2)$ if $n$ even and $n\ge 2$, and $SO_0(1,2)$ for $n=3$. Our proof is based on the Karpelevich Theorem and uses the classification of totally geodesic submanifolds of complex hyperbolic space and of the Lie ball. As an application we obtain a list of possible irreducible holonomy groups of Lorentzian conformal structures, namely $SO_0(2,n)$, SU(1,n), and $SO_0(1,2)$.

math.DG↗

arXiv2008

Riemannian geometry of Hartogs domains

Let $D_F = \{(z_0, z) \in {\C}^{n} | |z_0|^2 < b, \|z\|^2 < F(|z_0|^2) \}$ be a strongly pseudoconvex Hartogs domain endowed with the \K metric $g_F$ associated to the \K form $ω_F = -\frac{i}{2} \partial \bar{\partial} \log (F(|z_0|^2) - \|z\|^2)$. This paper contains several results on the Riemannian geometry of these domains. In the first one we prove that if $D_F$ admits a non special geodesic (see definition below) through the origin whose trace is a straight line then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of $D_F$ do not self-intersect, we find necessary and sufficient conditions on $F$ for $D_F$ to be geodesically complete and we prove that $D_F$ is locally irreducible as a Riemannian manifold. Finally, we compare the Bergman metric $g_B$ and the metric $g_F$ in a bounded Hartogs domain and we prove that if $g_B$ is a multiple of $g_F$, namely $g_B=λg_F$, for some $λ\in \R^+$, then $D_F$ is holomorphically isometric to an open subset of the complex hyperbolic space.

math.DG↗

arXiv2008

Symplectic duality between complex domains

In this paper after extending the definition of symplectic duality (given by the first two authors in arXiv:math/0603141 for bounded symmetric domains) to arbitrary complex domains of ${\C}^n$ centered at the origin we generalize some of the results proved in arXiv:math/0603141 and arXiv:0707.2125 to those domains.

math.SG↗

arXiv2008

Gray identities, canonical connection and integrability

We characterize quasi Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames introduced by the second author previously. By using such frames we also give a simpler and shorter proof of a Theorem of Goldberg. Furthermore we study almost Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

math.DG↗

arXiv2008

The bisymplectomorphism group of a bounded symmetric domain

An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in arXiv:math.DG/0603141 that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non compact symmetric spaces. In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is the direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma.

math.DG↗

arXiv2007

A geometric proof of the Karpelevich-Mostow's theorem

In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about existence of compatible Cartan decompositions.

math.DG↗

arXiv2006

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space.

math.DG↗

arXiv2006

Kahler manifolds and their relatives

Let $M_1$ and $M_2$ be two Kähler manifolds. We call $M_1$ and $M_2$ {\em relatives} if they share a non-trivial Kähler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) $h_1: S\to M_1$ and $h_2: S\to M_2$. Moreover, two Kähler manifolds $M_1$ and $M_2$ are said to be {\em weakly relatives} if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds $S_1$ and $S_2$ which admit two Kähler immersions into $M_1$ and $M_2$ respectively. The notions introduced are not equivalent. Our main results in this paper are Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex bounded domain with its Bergman metric and a projective Kähler manifold (i.e. a projective manifold endowed with the restriction of the Fubini--Study metric) are not relatives. In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective Kähler manifold are not weakly relatives. The proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature, i.e. no assumptions are used about the compactness or completeness of the manifolds involved. As a corollary we get that Hermitian symmetric spaces of different types are not weakly relatives.

math.DG↗

arXiv2005

Irreducibly acting subgroups of $Gl(n,\rr)$

In this note we prove the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup $G \subset Gl(n,\rr)$ is closed. Moreover, if $G$ admits an invariant bilinear form of Lorentzian signature, $G$ is maximal, i.e. it is conjugated to $SO(1,n-1)_0$. Finally we calculate the vector space of $G$-invariant symmetric bilinear forms, show that it is at most 3-dimensional, and determine the maximal stabilizers for each dimension.

math.DG↗

arXiv2001

Intrinsic palindromic numbers

We introduce a notion of palindromicity of a natural number which is independent of the base. We study the existence and density of palindromic and multiple palindromic numbers, and we raise several related questions.

math.GM↗