Abstract
In a recent paper [2] K. Nomizu has shown that a natural analogue of an n-sphere in an arbitrary Riemannian manifold is an n-dimensional umbilical submanifold with non-zero parallel mean curvature vector, which he calls “extrinsic sphere” sometimes. This note is concerned with the question whether extrinsic spheres have a special topological or differentiable feature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
KOBAYASHI, S. and NOMIZU, K.: Foundations of differential geometry, Volume II, New York, Wiley-Interscience, 1969.
NOMIZU, K.: Generalized central spheres and the notion of spheres in Riemannian geometry, Tôhoku Math. Journ.25, 129–137, (1973).
RECKZIEGEL, H.: Über Riemannsche Metriken, deren Werte, Levi-Civita-Zusammenhang und Riemannscher Krümmungstensor über einer Untermannigfaltigkeit vorgeschrieben sind, to appear in: Math. Ann. (1973).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Reckziegel, H. Submanifolds with prescribed mean curvature vector field. Manuscripta Math 13, 69–71 (1974). https://doi.org/10.1007/BF01168743
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01168743