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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces


Authors: Bang-Yen Chen, Franki Dillen, Leopold Verstraelen and Luc Vrancken
Journal: Proc. Amer. Math. Soc. 128 (2000), 589-598
MSC (1991): Primary 53B20; Secondary 53C42
Published electronically: July 23, 1999
MathSciNet review: 1664333
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Abstract: In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold $M$, denoted respectively by $\delta (n_{1},\ldots ,n_{k})$ and $\hat \delta (n_{1},\ldots ,n_{k})$, which trivially satisfy $\delta (n_{1},\ldots ,n_{k})\geq \hat \delta (n_{1},\ldots ,n_{k})$. In this article, we completely determine the Riemannian manifolds satisfying the condition $\delta (n_{1},\ldots ,n_{k})=\hat \delta (n_{1},\ldots ,n_{k})$. By applying the notions of these $\delta $-invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.


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Additional Information

Bang-Yen Chen
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: bychen@math.msu.edu

Franki Dillen
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: Franki.Dillen@wis.kuleuven.ac.be

Leopold Verstraelen
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: Leopold.Verstraelen@wis.kuleuven.ac.be

Luc Vrancken
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: Luc.Vrancken@wis.kuleuven.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05332-0
PII: S 0002-9939(99)05332-0
Keywords: Curvature, conformally flat space, Einstein space, $\delta $-invariants
Received by editor(s): April 17, 1997
Published electronically: July 23, 1999
Additional Notes: The second and fourth authors were supported by a postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium)(F.W.O.). Research supported by OT/TBA/95/9
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society