arXiv subjects

Massimo Vaccaro

Publications and source records attributed to Massimo Vaccaro.

4 recordsLinked to original sources

arXiv2022

$Sp(n)$-orbits in the Grassmannians of complex and $Σ$-complex subspaces of an Hermitian quaternionic vector space

We determine the invariants characterizing the $Sp(n)$-orbits in the real Grassmannian $Gr^\R(2k,4n)$ of the $2k$-dimensional complex and $Σ$-complex subspaces of a $4n$-dimensional Hermitian quaternionic vector space. A $Σ$-complex subspace is the orthogonal sum of complex subspaces by different, up to sign, compatible complex structure. The result is obtained by considering two main features of such subspaces. The first is that any such subspace admits a decomposition into an Hermitian orthogonal sum of 4-dimensional complex addends plus a 2-dimensional totally complex subspace if $k$ is odd, meaning that the quaternionification of the addends are orthogonal in pairs. The second is that any 4-dimensional complex addend $U$ is an isoclinic subspace i.e. the principal angles of the pair $(U,AU)$ are all the same for any compatible complex structure $A$. Using these properties we determine the full set of the invariants characterizing the $Sp(n)$-orbit of such subspaces in $Gr^\R(2k,4n)$.

math.DG↗

arXiv2021

$Sp(n)$-orbits of isoclinic subspaces in the real Grassmannians

In the framework of the study of the $Sp(n)$-orbits in the real Grassmannian $G^\R(k,4n)$ of $k$-dimensional non oriented subspaces of a real $4n$-dimensional vector space $V$, here we consider the case of the isoclinic subspaces whose set we indicate with $\mathcal{IC}$. Endowed $V$ with an Hermitian quaternionic structure $(\mathcal{Q},<,>)$, a subspace $U$ is isoclinic if for any compatible complex structure $A \in \mathcal{Q}$ the principal angles of the pair $(U,AU)$ are all the same, say $θ^A$. We will show that, fixed an admissible hypercomplex basis $(I,J,K)$, to any such subspace $U$ we can associate two set of invariants, namely a triple $(ξ,χ,η)$ and a pair $(Γ, Δ)$ where $Γ$ itself is a function of $(ξ,χ,η)$. We prove that the angles of isoclinicity $(θ^I,θ^J,θ^K)$ together with $(ξ,χ,η, Δ)$ determine its $Sp(n)$-orbit. In particular if $\dim U= 8k+2$ or $\dim U= 8k+6$ with $k \geq 0$ the last set reduce to the pair $(ξ= \pm 1,χ= \pm 1)$.

math.DG↗

arXiv2012

(Para-)Hermitian and (para-)Kähler Submanifolds of a para-quaternionic Kähler manifold

On a para-quaternionic Kähler manifold $(\widetilde M^{4n},Q,\widetilde g)$, which is first of all a pseudo-Riemannian manifold, a natural definition of (almost) Kähler and (almost) para-Kähler submanifold $(M^{2m},\mathcal{J},g)$ can be given where $\mathcal{J}=J_1|_M$ is a (para-)complex structure on $M$ which is the restriction of a section $J_1$ of the para-quaternionic bundle $Q$. In this paper, we extend to such a submanifold $M$ most of the results proved by Alekseevsky and Marchiafava, 2001, where Hermitian and Kähler submanifolds of a quaternionic Kähler manifold have been studied. Conditions for the integrability of an almost (para-)Hermitian structure on $M$ are given. Assuming that the scalar curvature of $\widetilde M$ is non zero, we show that any almost (para-)Kähler submanifold is (para-)Kähler and moreover that $M$ is (para-)Kähler iff it is totally (para-)complex. Considering totally (para-)complex submanifolds of maximal dimension $2n$, we identify the second fundamental form $h$ of $M$ with a tensor $C= J_2 \circ h \in TM\otimes S^2T^*M$ where $J_2 \in Q$ is a compatible para-complex structure anticommuting with $J_1$. When $\widetilde M^{4n}$ is a symmetric manifold the condition for a (para-)Kähler submanifold $M^{2n}$ to be locally symmetric is given. In the case when $\widetilde M$ is a para-quaternionic space form, it is shown, by using Gauss and Ricci equations, that a (para-)Kähler submanifold $M^{2n}$ is curvature invariant. Moreover it is a locally symmetric Hermitian submanifold iff the $\mathfrak u(n)$-valued 2-form $[C,C]$ is parallel. %$$[C,C]: X\wedge Y \mapsto [C_X,C_Y], \qquad \quad X,Y \in TM$$ %(which satisfies the first and the second Bianchi identity) is parallel. Finally a characterization of \textit{parallel} Kähler and para-Kähler submanifold of maximal dimension is given.

math.DG↗

arXiv2010

Subspaces of a para-quaternionic Hermitian vector space

Let $(\tilde Q,g) $ be a para-quaternionic Hermitian structure on the real vector space $V$. By referring to the tensorial presentation $(V, \tilde{Q},g) \simeq (H^2 \otimes E^{2n}, \mathfrak{sl}(H),ω^H \otimes ω^E)$, we give an explicit description, from an affine and metric point of view, of main classes of subspaces of $V$ which are invariantly defined with respect to the structure group of $\tilde{Q}$ and $(\tilde{Q},g)$ respectively.

math.DG↗