## Cylindricity of isometric immersions into Euclidean space

HTML articles powered by AMS MathViewer

- by Robert Maltz
- Proc. Amer. Math. Soc.
**53**(1975), 428-432 - DOI: https://doi.org/10.1090/S0002-9939-1975-0643658-X
- PDF | Request permission

## Abstract:

A simple geometric proof is given for the Hartman-Nirenberg cylindricity theorem and some generalizations. Then the following cylindricity theorem (unpublished) of S. Alexander is proved using the same idea. Theorem.*Let*$f:M \to {E^n}$

*be an isometric Euclidean immersion of the Riemannian product*$M = {M_1} \times \ldots \times {M_k} \times {E^m}$

*where the*${M_i}$

*are not everywhere flat Riemannian manifolds, and*${E^m}$

*denotes Euclidean*$m$-

*space. Then*$C \geqslant k$,

*where*$C$

*denotes the codimension of the immersion; and if*$C = k$,

*then the immersion is cylindrical on the Euclidean factor*.

## References

- S. Alexander and R. Maltz,
*Isometric immersions of Riemannian products in Euclidean space*, J. Differential Geometry**11**(1976), no. 1, 47–57. MR**417990** - Jeff Cheeger and Detlef Gromoll,
*The splitting theorem for manifolds of nonnegative Ricci curvature*, J. Differential Geometry**6**(1971/72), 119–128. MR**303460** - Shiing-shen Chern and Nicolaas H. Kuiper,
*Some theorems on the isometric imbedding of compact Riemann manifolds in euclidean space*, Ann. of Math. (2)**56**(1952), 422–430. MR**50962**, DOI 10.2307/1969650 - Philip Hartman,
*On the isometric immersions in Euclidean space of manifolds with nonnegative sectional curvatures. II*, Trans. Amer. Math. Soc.**147**(1970), 529–540. MR**262981**, DOI 10.1090/S0002-9947-1970-0262981-4 - Philip Hartman and Louis Nirenberg,
*On spherical image maps whose Jacobians do not change sign*, Amer. J. Math.**81**(1959), 901–920. MR**126812**, DOI 10.2307/2372995 - André Lichnerowicz,
*Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative*, J. Differential Geometry**6**(1971/72), 47–94 (French). MR**300228** - William S. Massey,
*Surfaces of Gaussian curvature zero in Euclidean $3$-space*, Tohoku Math. J. (2)**14**(1962), 73–79. MR**139088**, DOI 10.2748/tmj/1178244205 - John Douglas Moore,
*Isometric immersions of riemannian products*, J. Differential Geometry**5**(1971), 159–168. MR**307128** - Katsumi Nomizu,
*On hypersurfaces satisfying a certain condition on the curvature tensor*, Tohoku Math. J. (2)**20**(1968), 46–59. MR**226549**, DOI 10.2748/tmj/1178243217 - A. V. Pogorelov,
*Extensions of the theorem of Gauss on spherical representation to the case of surfaces of bounded extrinsic curvature*, Dokl. Akad. Nauk SSSR (N.S.)**111**(1956), 945–947 (Russian). MR**0087147**

## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**53**(1975), 428-432 - MSC: Primary 53C40
- DOI: https://doi.org/10.1090/S0002-9939-1975-0643658-X
- MathSciNet review: 0643658