Codazzi Tensors and Reducible Submanifolds

Bivens, Irl C.

doi:10.2307/1998347

Cited by 2 publications

(1 citation statement)

References 4 publications

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“…(b) Suppose/? = 1, M has constant sectional curvature and N is a unit normal vector field on M. The shape operator T of M is a tensor of type (1,1) on M defined by T(X) --VXN where X denotes a tangent vector to M. Since M has constant sectional curvature the Codazzi equations for M imply T is a Codazzi tensor of type (1,1). As a consequence S=Tk=T*T* ■•• * T (k times) is a Codazzi tensor of type (k, k) and trace S = k\("k)ak (the factor of k\ occurs because of our particular normalization of wedge products).…”

Section: Integral Formulas and Hyperspheres In A Simply Connected Spamentioning

confidence: 99%

Integral formulas and hyperspheres in a simply connected space form

Bivens¹

1983

Proc. Amer. Math. Soc.

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Abstract.Let M" denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic n + 1 space or in an open hemisphere of S"+ '. We show that if two consecutive mean curvatures of M are constant then M is in fact a geodesic sphere. The proof uses the generalized Minkowski integral formulas for a hypersurface of a complete simply connected space form. These Minkowski formulas are derived from an integral formula for submanifolds in which the ambient Riemannian manifold M possesses a generalized position vector field; that is a vector field Y whose covariant derivative is at each point a multiple of the identity. In addition we prove that if M is complete and connected with the covariant derivative of Y exactly the identity at each point then M is isometric to Euclidean space.

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Section: Integral Formulas and Hyperspheres In A Simply Connected Spamentioning

confidence: 99%

Integral formulas and hyperspheres in a simply connected space form

Bivens¹

1983

Proc. Amer. Math. Soc.

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Orthogonal geodesic and minimal distributions

Bivens¹

1983

Trans. Amer. Math. Soc.

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Abstract. Let 8 be a smooth distribution on a Riemannian manifold M with $ the orthogonal distribution. We say that 5 is geodesic provided g is integrable with leaves which are totally geodesic submanifolds of M. The notion of minimality of a submanifold of M may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to ip then we say ip is minimal. Suppose that 8 and § are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of 3 is also a submanifold of Euclidean space with mean curvature normal vector field r¡. We show that the integral of | ij |2 over M is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold.We study the relationships between the geometry of M and the integrability of §. For example, if 3 and § are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then § is integrable iff 3 and § are parallel distributions. Similarly if M" has constant negative sectional curvature and dim § = 2 < n then § is not integrable. If 3 is geodesic and § is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of § are flat submanifolds of A/, with parallel second fundamental form.

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Codazzi Tensors and Reducible Submanifolds

Cited by 2 publications

References 4 publications

Integral formulas and hyperspheres in a simply connected space form

Integral formulas and hyperspheres in a simply connected space form

Orthogonal geodesic and minimal distributions

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