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Centroaffine immersions of codimension two and projective hypersurface theory

Published online by Cambridge University Press:  22 January 2016

Katsumi Nomizu  and
Takeshi Sasaki
Katsumi Nomizu
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912USA
Takeshi Sasaki
Affiliation:
Department of Mathematics, Kobe University, Rokko, Kobe 657Japan
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Affine differential geometry developed by Blaschke and his school [B] has been reorganized in the last several years as geometry of affine immersions. An immersion f of an n-dimensional manifold M with an affine connection ∇ into an (n + 1)-dimensional manifold Ḿwith an affine connection ∇ is called an affine immersion if there is a transversal vector field ξ such that ∇xf*(Y) = f*(∇xY) + h(X,Y)ξ holds for any vector fields X, Y on Mn. When f: Mn→ Rn+1 is a nondegenerate hypersurface, there is a uniquely determined transversal vector field ξ, called the affine normal field, an essential starting point in classical affine differential geometry. The new point of view allows us to relax the non-degeneracy condition and gives us more freedom in choosing ξ; what this new viewpoint can accomplish in relating affine differential geometry to Riemannian geometry and projective differential geometry can be seen from [NP1], [NP2], [NS] and others. For the definitions and basic formulas on affine immersions, centroaffine immersions, conormal (or dual) maps, projective flatness, etc., the reader is referred to [NP1]. These notions will be generalized to codimension 2 in this paper.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 132 , December 1993 , pp. 63 - 90
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

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