Totally geodesic submanifolds of regular Sasakian manifolds
type: Item
instance of: scholarly article
title: Totally geodesic submanifolds of regular Sasakian manifolds
zbMATH Open document ID: 1244.53063
published in: Osaka Journal of Mathematics
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1332337241
review text: The main result of this paper finds a correspondence between certain types of totally geodesic submanifolds of a homogeneous Sasakian manifold (necessarily regular) and submanifolds of its Hermitian symmetric base space (via the Boothby-Wang fibration). Namely, invariant submanifolds (with respect to the \((1,1)\) tensor of the Sasakian structure) correspond to complex submanifolds of the base, and anti-invariant submanifolds of maximal dimension (necessarily Legendrian with respect to the underlying contact structure) correspond to totally real submanifolds of the base. As a consequence, there are no totally geodesic, contact CR submanifolds in a homogeneous Sasakian manifold. This results extend and complete earlier ones by Yano \& Kon, Leung etc. On the other hand, the result can be seen as a step toward the classification of totally geodesic submanifolds of Sasakian manifolds and as a recipe for producing examples. The proof relies on local computations. The paper contains a very useful up to date presentation of the progress in the problem of classifying totally geodesic submanifolds.
reviewed by: Liviu Ornea
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 6029030
zbMATH Keywords: Sasakian manifold
zbMATH Keywords: homogeneous space
zbMATH Keywords: totally geodesic submanifold
zbMATH Keywords: invariant submanifold
zbMATH Keywords: antiinvariant submanifold
MaRDI profile type: Publication
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author: Thomas Murphy
description: scientific article; zbMATH DE number 6029030