On linear holonomy group of Riemannian symmetric spaces

Riemannian Twistor Spaces and Holonomy Groups

• The Penrose fibration CP 3 → HP 1 = S4 encodes a large part of the geometry of the 4-sphere. For instance, instanton solutions to the Yang-Mills equations on S4 pull back to holomorphic bundles on CP 3 [30]. This forms the basis of the Atiyah-Drinfeld-HitchinManin classification of the instantons [1]. Again, horizontal curves in CP 3 project onto branched minimal surfaces in S4. Using this Br...

Symmetric Submanifolds of Riemannian Symmetric Spaces

A symmetric space is a Riemannian manifold that is “symmetric” about each of its points: for each p ∈M there is an isometry σp of M such that (σp)∗ = −I on TpM . Symmetric spaces and their local versions were studied and classified by E.Cartan in the 1920’s. In 1980 D.Ferus [F2] introduced the concept of symmetric submanifolds of Euclidean space: A submanifold M of R is a symmetric submanifold ...

Discrete Groups, Symmetric Spaces, and Global Holonomy

Conformal holonomy, symmetric spaces, and skew symmetric torsion

We consider the question: can the isotropy representation of an irreducible pseudoRiemannian symmetric space be realized as a conformal holonomy group? Using recent results of Čap, Gover and Hammerl, we study the representations of SO(2, 1), PSU(2, 1) and PSp(2, 1) as isotropy groups of irreducible symmetric spaces of signature (3, 2), (4, 4) and (6, 8), respectively, describing the geometry in...

Involutions of compact Riemannian 4-symmetric spaces

Let G/H be a compact 4-symmetric space of inner type such that the dimension of the center Z(H) of H is at most one. In this paper we shall classify involutions of G preserving H for the case where dimZ(H) = 0, or H is a centralizer of a toral subgroup of G.