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Alberto Dolcetti

Publications and source records attributed to Alberto Dolcetti.

10 recordsLinked to original sources
arXiv2024

Some Riemannian properties of $\mathbf{SU_n}$ endowed with a bi-invariant metric

We study some properties of $SU_n$ endowed with the Frobenius metric $\phi$, which is, up to a positive constant multiple, the unique bi-invariant Riemannian metric on $SU_n$. In particular we express the distance between $P, Q \in SU_n$ in terms of eigenvalues of $P^*Q$; we compute the diameter of $(SU_n, \phi)$ and we determine its diametral pairs; we prove that the set of all minimizing geodesic segments with endpoints $P$, $Q$ can be parametrized by means of a compact connected submanifold of $\mathfrak{su}_n$, diffeomorphic to a suitable complex Grassmannian depending on $P$ and $Q$.

math.DG↗
arXiv2022

SVD-closed subgroups of the unitary group: generalized principal logarithms and minimizing geodesics

We study the set of generalized principal $\mathfrak{g}$-logarithms of any matrix belonging to a connected SVD-closed subgroup $G$ of $U_n$, with Lie algebra $\mathfrak{g}$. This set is a non-empty disjoint union of a finite number of subsets diffeomorphic to homogeneous spaces, and it is related to a suitable set of minimizing geodesics. Many particular cases for the group $G$ are explicitly analysed.

math.DG↗
arXiv2017

Some remarks on the Jordan-Chevalley decomposition

In this note we mainly study the fine Jordan-Chevalley decomposition: a refinement of the classical Jordan-Chevalley decomposition of a matrix and we pay a particular attention to the field of the coefficients of the matrix. Moreover we obtain some further additive and multiplicative decompositions of a matrix under suitable conditions.

math.RA↗
arXiv2016

Skew symmetric logarithms and geodesics on $O_n(\RR)$

We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.

math.DG↗
arXiv2014

Some differential properties of $GL_n(\mathbb{R})$ with the trace metric

In this note we consider some properties of $GL_n(\mathbb{R})$ with the Semi-Riemannian structure induced by the trace metric $g$. In particular we study geodesics and curvature tensors. Moreover we prove that $GL_n$ has a suitable foliation, whose leaves are isometric to $(SL_n(\mathbb{R}), g)$, while its component of matrices with positive determinant is isometric to the Semi-Riemannian product manifold $SL_n \times \mathbb{R}$.

math.DG↗