Geometry of tubes about characteristic curves on Sasakian manifolds
type: Item
instance of: scholarly article
title: Geometry of tubes about characteristic curves on Sasakian manifolds
zbMATH Open document ID: 0755.53028
DOI: 10.1007/BF02844468
author: Lieven Vanhecke
author: Mirjana Djoric
publication date: 1993-01-16T00:00:00Z
review text: Let \(M\) be a Sasakian manifold with contact form \(\eta\), characteristic vector field \(\xi\) and fundamental collineation \(\varphi\). If the sectional curvatures of plane sections spanned by a vector \(X\) orthogonal to \(\xi\) and \(\varphi X\) are constant, \(M\) is called a Sasakian space form (or said to be of constant \(\varphi\)-sectional curvature). If \(\varphi^ 2(\nabla_ VR)_{XY}Z=0\) for \(V\), \(X\), \(Y\), \(Z\) orthogonal to \(\xi\), \(M\) is said to be locally \(\varphi\)-symmetric, [\textit{J. Takahashi}, Tohoku Math. J., II. Ser. 29, 91-113 (1977; Zbl 0343.53030)]. Let \(\sigma\) be an integral curve of \(\xi\) and \(P_ r\) the geodesic tube about \(\sigma\) of radius \(r\). Finally for \(p \in P_ r\), put \(p=\exp_ m(ru)\), \(m \in \sigma\); where \(u\) is a unit normal to \(\sigma\) at \(m\) and denote by \(\kappa_ \sigma\) the curvature at \(p\) of the geodesic of \(P_ r\) tangent to \(\varphi u\). The authors prove that a Sasakian manifold of dimension \(\geq 5\) is a Sasakian space form if and only if the integral curves of \(\varphi u\) on each \(P_ r\) are geodesics if and only if \(\varphi u\) is a Killing field on each \(P_ r\). It is also shown that a Sasakian manifold of dimension \(\geq 5\) is locally \(\varphi\)-symmetric if and only if for all \(u\) (as above) and all small \(r\), \(\kappa_ \sigma(\exp_ \sigma(ru))=\kappa_ \sigma(\exp_ \sigma(-ru))\). Similar characterizations are also obtained using the Ricci operator of the geodesic tubes instead of the shape operator. Proofs make use of the fact that \(\xi\) induces a local fibration over a Kähler manifold \(B\) and that \(M\) is a Sasakian space form if and only if \(B\) is a complex space form, respectively that \(M\) is locally \(\varphi\)-symmetric if and only if \(B\) is locally Hermitian symmetric, one of the main results of the Takahashi paper referred to above.
reviewed by: David E. Blair
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 91671
zbMATH Keywords: \(\varphi\)-symmetric space
zbMATH Keywords: Sasakian space form
zbMATH Keywords: geodesic tubes
MaRDI profile type: Publication
cites work: Contact manifolds in Riemannian geometry
cites work: Symmetries and \(\phi\)-symmetric spaces
cites work: Q4010808
cites work: Q3982336
cites work: Q3134786
cites work: Q3973826
cites work: The Volumes of Tubes about Curves in a Riemannian Manifold
cites work: Q3993500
cites work: A study on certain homogeneous spaces
cites work: Q4731791
cites work: On fiberings of almost contact manifolds
cites work: Sasakian \(\varphi\)-symmetric spaces
cites work: Transversally symmetric Riemannian foliations
cites work: Interaction of tubes and spheres
cites work: Q3222791
Wikidata QID: Q125366735
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 91671