NDLI: Codazzi tensors and reducible submanifolds

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Codazzi tensors and reducible submanifolds

Content Provider	Semantic Scholar
Author	Bivens, Irl C.
Copyright Year	1981
Abstract	An integral formula is derived for Codazzi tensors of type (k, k). Many of the classical Minkowski type integral formulas then become special cases of this one. If M is a submanifold of Euclidean space and it is a parallel distribution on M then each leaf of m is a submanifold of Euclidean space with mean curvature normal vector field tj. Using the above integral formula we show that the integral of \|tj\|2 over M is bounded below by an intrinsic constant and we give necessary and sufficient conditions for equality to hold. The reducible surfaces for which equality holds are characterized and related results concerned with Riemannian product manifolds are proved. Parallel tensors of type (1,1) are characterized in terms of the de Rham decomposition. It is shown that if M is irreducible and A is a parallel tensor of type (1, 1) on M which is not multiplication by a constant then M is a Kaehler manifold. Some further results are derived for manifolds whose simply connected cover is Kaehler. Acknowledgements. Most of the results of this paper were contained in my thesis written under the direction of Professor Patrick Eberlein at the University of North Carolina at Chapel Hill. It is a pleasure to acknowledge here my debt to Professor Eberlein, both for his guidance in matters mathematical, and for his unfailing support and encouragement.
Starting Page	231
Ending Page	246
Page Count	16
File Format	PDF HTM / HTML
DOI	10.1090/S0002-9947-1981-0628456-2
Volume Number	268
Alternate Webpage(s)	https://www.ams.org/journals/tran/1981-268-01/S0002-9947-1981-0628456-2/S0002-9947-1981-0628456-2.pdf
Alternate Webpage(s)	http://www.ams.org/journals/tran/1981-268-01/S0002-9947-1981-0628456-2/S0002-9947-1981-0628456-2.pdf
Alternate Webpage(s)	https://doi.org/10.1090/S0002-9947-1981-0628456-2
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article

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