*Bull. Austral. Math. Soc.* 72(3) pp.391--402, 2005.

# Examples and classification of

Riemannian submersions satisfying a basic equality

## Bang-Yen Chen |

.

## Abstract

In an earlier article we obtain a sharp
inequality for an arbitrary isometric immersion from a Riemannian
manifold admitting a Riemannian submersion with totally geodesic
fibres into a unit sphere. In this article we investigate the
immersions which satisfy the equality case of the inequality. As a
by-product, we discover a new characterisation of Cartan
hypersurface in

*S*^{4}.#### Click to download PDF of this article (free access until July 2006)

#### or get the *no-frills* version

[an error occurred while processing this directive]
(Metadata: XML, RSS, BibTeX) | MathSciNet: MR2199641 | Z'blatt-MATH: 1093.53064 |

## References

- D.E. Blair;

*Riemannian geometry of contact and symplectic manifolds*(BirkhĂ¤user Boston, Inc., Boston, MA, 2002).**MR1874240** - B.Y. Chen;

*Geometry of submanifolds*(Mercel Dekker, New York, 1973).**MR353212** - B.Y. Chen;

Totally umbilical submanifolds of quaternion-space-forms,

*J. Austral. Math. Soc. Ser. A***26**(1978), pp. 154--162.**MR511599** - B.Y. Chen;

Some pinching and classification theorems for minimal submanifolds,

*Arch. Math.***60**(1993), pp. 568--578.**MR1216703** - B.Y. Chen;

Some new obstructions to minimal and Lagrangian isometric immersions,

*Japan. J. Math.***26**(2000), pp. 105--127.**MR1771434** - B.Y. Chen;

Riemannian submersions, minimal immersions and cohomology class,

(submitted). - R.H. Escobales, Jr.;

Riemannian submersions with totally geodesic fibers,

*J. Differential Geom.***10**(1975), pp. 253--276.**MR370423** - R.H. Escobales, Jr.;

Riemannian submersions from complex projective space,

*J. Differential Geom.***13**(1978), pp. 93--107.**MR520604** - W.Y. Hsiang and H.B. Lawson, Jr.;

Minimal submanifolds of low cohomogeneity,

*J. Differential Geom.***5**(1971), pp. 1--38.**MR298593** - T. Nagano;

On fibred Riemann manifolds,

*Sci. Papers College Gen. Ed. Univ. Tokyo***10**(1960), pp. 17--27.**MR157325** - B. O'Neill;

The fundamental equations of a submersion,

*Michigan Math. J.***13**(1966), pp. 459--469.**MR200865**

ISSN 0004-9727