Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map
Identifieur interne : 000A51 ( Istex/Checkpoint ); précédent : 000A50; suivant : 000A52Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map
Auteurs : Tillmann Jentsch [Allemagne] ; Helmut Reckziegel [Allemagne]Source :
- Annals of Global Analysis and Geometry [ 0232-704X ] ; 2006-01-01.
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Abstract
Abstract: For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative $$\hat{\nabla}$$ on the Grassmann bundle ^ := Gm(T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rn this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ Gm(TpN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving $$\hat{\nabla}$$ ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.
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DOI: 10.1007/s10455-006-1146-7
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(For N Rn this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ Gm(TpN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving $$\hat{\nabla}$$ ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.</div></front></TEI><affiliations><list><country><li>Allemagne</li></country></list><tree><country name="Allemagne"><noRegion><name sortKey="Jentsch, Tillmann" sort="Jentsch, Tillmann" uniqKey="Jentsch T" first="Tillmann" last="Jentsch">Tillmann Jentsch</name></noRegion><name sortKey="Jentsch, Tillmann" sort="Jentsch, Tillmann" uniqKey="Jentsch T" first="Tillmann" last="Jentsch">Tillmann Jentsch</name><name sortKey="Reckziegel, Helmut" sort="Reckziegel, Helmut" uniqKey="Reckziegel H" first="Helmut" last="Reckziegel">Helmut Reckziegel</name><name sortKey="Reckziegel, Helmut" sort="Reckziegel, Helmut" uniqKey="Reckziegel H" first="Helmut" last="Reckziegel">Helmut Reckziegel</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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