15,995 research outputs found
Naturally reductive pseudo-Riemannian spaces
A family of naturally reductive pseudo-Riemannian spaces is constructed out
of the representations of Lie algebras with ad-invariant metrics. We exhibit
peculiar examples, study their geometry and characterize the corresponding
naturally reductive homogeneous structure.Comment: A shorter, clearer and more concise versio
Riemannian Holonomy Groups of Statistical Manifolds
Normal distribution manifolds play essential roles in the theory of
information geometry, so do holonomy groups in classification of Riemannian
manifolds. After some necessary preliminaries on information geometry and
holonomy groups, it is presented that the corresponding Riemannian holonomy
group of the -dimensional normal distribution is
, for all . As a
generalization on exponential family, a list of holonomy groups follows.Comment: 11 page
A new construction of homogeneous quaternionic manifolds and related geometric structures
Let V be the pseudo-Euclidean vector space of signature (p,q), p>2 and W a
module over the even Clifford algebra Cl^0 (V). A homogeneous quaternionic
manifold (M,Q) is constructed for any spin(V)-equivariant linear map \Pi :
\wedge^2 W \to V. If the skew symmetric vector valued bilinear form \Pi is
nondegenerate then (M,Q) is endowed with a canonical pseudo-Riemannian metric g
such that (M,Q,g) is a homogeneous quaternionic pseudo-K\"ahler manifold.
The construction is shown to have a natural mirror in the category of
supermanifolds. In fact, for any spin(V)-equivariant linear map \Pi : Sym^2 W
\to V a homogeneous quaternionic supermanifold (M,Q) is constructed and,
moreover, a homogeneous quaternionic pseudo-K\"ahler supermanifold (M,Q,g) if
the symmetric vector valued bilinear form \Pi is nondegenerate.Comment: to appear in the Memoirs of the AMS, 81 pages, Latex source fil
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