Universität zu Köln

Submanifolds with parallel second fundamental form studied via the Gauß map

Jentsch, Tillmann (2006) Submanifolds with parallel second fundamental form studied via the Gauß map. PhD thesis, Universität zu Köln.

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    Abstract

    For an arbitrary n-dimensional riemannian manifold N and an integer m between 1 and n-1 a covariant derivative on the Graßmann bundle G_m(TN) is introduced which has the property that an m-dimensional submanifold M of N has parallel second fundamental form if and only if its Gauß map (defined on M with values in G_m(TN)) is affine. (For the case that N is the euclidian space this result was already obtained by J.Vilms in 1972.) By means of this relation a generalization of E. Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing an m-dimensional tangent space W and a second fundamental form b at p in N are given; for these data we construct an m-dimensional ``umbrella'' M=M(p,W,b) in N, the rays of which are helical arcs of N; moreover we present tensorial conditions (not involving the covariant derivative on G_m(TN)) 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.

    Item Type: Thesis (PhD thesis)
    Creators:
    CreatorsEmail
    Jentsch, Tillmanntjentsch@math.uni-koeln.de
    URN: urn:nbn:de:hbz:38-17915
    Subjects: Mathematics
    Uncontrolled Keywords:
    KeywordsLanguage
    parallel submanifold , Graßmann bundle , Gauß mapEnglish
    Faculty: Mathematisch-Naturwissenschaftliche Fakultät
    Divisions: Mathematisch-Naturwissenschaftliche Fakultät > Mathematisches Institut
    Language: English
    Date: 2006
    Date Type: Completion
    Date of oral exam: 11 July 2005
    Full Text Status: Public
    Date Deposited: 19 Jul 2006 13:16:52
    Referee
    NameAcademic Title
    Reckziegel, H.Prof. Dr.
    URI: http://kups.ub.uni-koeln.de/id/eprint/1791

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