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On the RICCI Metric of a Hypersurface

โœ Scribed by Thomas Hasanis

Publisher
John Wiley and Sons
Year
1991
Tongue
English
Weight
422 KB
Volume
154
Category
Article
ISSN
0025-584X

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โœฆ Synopsis

The last years many results have been published about the existence of a RIEMANNian metric on a differentiable manifold, with prescribed RICCI tensor. However, if the RIEMANNian manifold (M, (,)) has positive RICCI curvature, then the RICCI tensor defines a new RIEMANNian metric on M, which is denoted by (,)* and is called the RICCI metric. We study this metric in the case of ovaloids of EucLIDean spaces. Among other results it is proved that max Ric 2 1, where Ric denotes the RICCI curvature of the RICCI metric. This gives a further necessary condition for (M, g) to be realized as an ovaloid in En+', with g as RICCI tensor. * * * *

2. Conformal metrics

In the EucLIDean space E"" we consider a hypersurface M with induced metric (,). Let (,) = f( ,) be a conformal metric, where f: M 4 R is a positive smooth function on M . -* * * * * = (VxVrY, X>* -(VrVxY, X>* -(V[x,r]K X>*

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