Codazzi tensors and reducible submanifolds

Author:
Irl Bivens

Journal:
Trans. Amer. Math. Soc. **268** (1981), 231-246

MSC:
Primary 53C40

DOI:
https://doi.org/10.1090/S0002-9947-1981-0628456-2

MathSciNet review:
628456

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\pi$ is a parallel distribution on $M$ then each leaf of $\pi$ is a submanifold of Euclidean space with mean curvature normal vector field $\eta$. Using the above integral formula we show that the integral of ${\left | \eta \right |^2}$ 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.

- Marcel Berger, Paul Gauduchon, and Edmond Mazet,
*Le spectre d’une variété riemannienne*, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR**0282313** - Irl Bivens,
*Codazzi tensors and reducible submanifolds*, Trans. Amer. Math. Soc.**268**(1981), no. 1, 231–246. MR**628456**, DOI https://doi.org/10.1090/S0002-9947-1981-0628456-2 - Bang-yen Chen and Kentaro Yano,
*Integral formulas for submanifolds and their applications*, J. Differential Geometry**5**(1971), 467–477. MR**288699** - Robert B. Gardner,
*The technique of integral formulas in the geometry of immersions*, Differentialgeometrie im Grossen (Tagung, Math. Forschungsinst., Oberwolfach, 1969) Bibliographisches Inst., Mannheim, 1971, pp. 127–144. Ber. Math. Forschungsinst. Oberwolfach, No. 4. MR**0358638** - Robert B. Gardner,
*New viewpoints in the geometry of submanifolds of ${\bf R}^{N}$*, Bull. Amer. Math. Soc.**83**(1977), no. 1, 1–35. MR**431044**, DOI https://doi.org/10.1090/S0002-9904-1977-14174-8 - Samuel I. Goldberg and Kentaro Yano,
*Globally framed $f$-manifolds*, Illinois J. Math.**15**(1971), 456–474. MR**278247** - Alfred Gray,
*A note on manifolds whose holonomy group is a subgroup of ${\rm Sp}(n)\cdot {\rm Sp}(1)$*, Michigan Math. J.**16**(1969), 125–128. MR**244913** - Chuan-Chih Hsiung,
*Some integral formulas for closed hypersurfaces*, Math. Scand.**2**(1954), 286–294. MR**68236**, DOI https://doi.org/10.7146/math.scand.a-10415 - Shoshichi Kobayashi,
*On compact Kähler manifolds with positive definite Ricci tensor*, Ann. of Math. (2)**74**(1961), 570–574. MR**133086**, DOI https://doi.org/10.2307/1970298 - Shoshichi Kobayashi and Katsumi Nomizu,
*On automorphisms of a Kählerian structure*, Nagoya Math. J.**11**(1957), 115–124. MR**97536**
---, - John Douglas Moore,
*Isometric immersions of riemannian products*, J. Differential Geometry**5**(1971), 159–168. MR**307128** - Robert C. Reilly,
*On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space*, Comment. Math. Helv.**52**(1977), no. 4, 525–533. MR**482597**, DOI https://doi.org/10.1007/BF02567385

*Foundations of differential geometry*, Interscience, New York, 1963.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53C40

Retrieve articles in all journals with MSC: 53C40

Additional Information

Article copyright:
© Copyright 1981
American Mathematical Society