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# Integral formulas in Riemannian geometry

01 Jan 1970-

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01 Jan 1990

TL;DR: The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990.

Abstract: The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.

258 citations

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TL;DR: The hypersurfaces of a Euclidean space have been studied and the existence of (QE)n manifolds is proved.

Abstract: In this paper we give some examples of a quasi Einstein manifold (QE)n. Next we prove the existence of (QE)n manifolds. Then we study some properties of a quasi Einstein manifold. Finally the hypersurfaces of a Euclidean space have been studied.

194 citations

### Cites background from "Integral formulas in Riemannian geo..."

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TL;DR: In this paper, Chen and Nagano established a general method to determine stability of totally geodesic submanifolds of symmetric spaces, and established a stability theorem for minimal totally real sub- manifolds of Kahlerian manifolds.

Abstract: One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds M+,M introduced and studied by Chen and Nagano in (Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405-425) as minimal submanifolds. The other purpose is to establish a stability theorem for minimal totally real sub- manifolds of Kahlerian manifolds.

184 citations

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TL;DR: In this paper, the geometric properties of biharmonic curves and surfaces of some Thurston's geometries have been discussed, including the biharmonicity of maps between warped products.

Abstract: points of the bienergy functional E2(’) = 1 R M j?(’)j 2 vg; where ?(’) is the tension fleld of ’. Biharmonic maps are a natural expansion of harmonic maps (?(’) = 0). Although E2 has been on the mathematical scene since the early ’60, when some of its analytical aspects have been discussed, and regularity of its critical points is nowadays a well-developed fleld, a systematic study of the geometry of biharmonic maps has started only recently. In this lecture we focus on the geometric properties of biharmonic maps and describe some recent achievements on the subject: (a) We give the explicit classiflcations of biharmonic curves and surfaces of some Thurston’s geometries [2, 3, 4]. (b) We describe the biharmonicity of maps between warped products and using this setting we study three classes of axially symmetric biharmonic maps [1]. (c) Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy, show it derives from a variational problem on metrics, exhibit the peculiarity of dimension four, and use the stress-energy tensor to construct new examples of biharmonic maps [5].

145 citations

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01 Jan 1990

TL;DR: The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990.

Abstract: The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.

258 citations

••

[...]

TL;DR: The hypersurfaces of a Euclidean space have been studied and the existence of (QE)n manifolds is proved.

Abstract: In this paper we give some examples of a quasi Einstein manifold (QE)n. Next we prove the existence of (QE)n manifolds. Then we study some properties of a quasi Einstein manifold. Finally the hypersurfaces of a Euclidean space have been studied.

194 citations

••

[...]

TL;DR: In this paper, Chen and Nagano established a general method to determine stability of totally geodesic submanifolds of symmetric spaces, and established a stability theorem for minimal totally real sub- manifolds of Kahlerian manifolds.

Abstract: One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds M+,M introduced and studied by Chen and Nagano in (Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405-425) as minimal submanifolds. The other purpose is to establish a stability theorem for minimal totally real sub- manifolds of Kahlerian manifolds.

184 citations

[...]

TL;DR: In this paper, the geometric properties of biharmonic curves and surfaces of some Thurston's geometries have been discussed, including the biharmonicity of maps between warped products.

Abstract: points of the bienergy functional E2(’) = 1 R M j?(’)j 2 vg; where ?(’) is the tension fleld of ’. Biharmonic maps are a natural expansion of harmonic maps (?(’) = 0). Although E2 has been on the mathematical scene since the early ’60, when some of its analytical aspects have been discussed, and regularity of its critical points is nowadays a well-developed fleld, a systematic study of the geometry of biharmonic maps has started only recently. In this lecture we focus on the geometric properties of biharmonic maps and describe some recent achievements on the subject: (a) We give the explicit classiflcations of biharmonic curves and surfaces of some Thurston’s geometries [2, 3, 4]. (b) We describe the biharmonicity of maps between warped products and using this setting we study three classes of axially symmetric biharmonic maps [1]. (c) Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy, show it derives from a variational problem on metrics, exhibit the peculiarity of dimension four, and use the stress-energy tensor to construct new examples of biharmonic maps [5].

145 citations

##### Related Papers (5)

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