Codazzi structures induced by minimal affine immersions

Cubic form Theorem for Affine Immersions

Results in Mathematics ◽

10.1007/bf03323250 ◽

1988 ◽

Vol 13 (3-4) ◽

pp. 338-362 ◽

Cited By ~ 19

Author(s):

Katsumi Nomizu ◽

Ulrich Pinkall

Keyword(s):

Cubic Form ◽

Form Theorem ◽

Affine Immersions

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An existence theorem for $G$-structure preserving affine immersions

Indiana University Mathematics Journal ◽

10.1512/iumj.2008.57.3281 ◽

2008 ◽

Vol 57 (3) ◽

pp. 1431-1465 ◽

Cited By ~ 9

Author(s):

Paolo Piccione ◽

Daniel V. Tausk

Keyword(s):

Existence Theorem ◽

Structure Preserving ◽

Affine Immersions

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Centroaffine immersions of codimension two and projective hypersurface theory

Nagoya Mathematical Journal ◽

10.1017/s0027763000004645 ◽

1993 ◽

Vol 132 ◽

pp. 63-90 ◽

Cited By ~ 14

Author(s):

Katsumi Nomizu ◽

Takeshi Sasaki

Keyword(s):

Differential Geometry ◽

Vector Field ◽

Affine Connection ◽

Affine Differential Geometry ◽

Dimensional Manifold ◽

Affine Immersion ◽

Projective Hypersurface ◽

Starting Point ◽

Affine Immersions ◽

Transversal Vector Field

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.

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