ON FIBRATIONS WITH FLAT FIBERS

Eppstein David

Description

ON FIBRATIONS WITH FLAT FIBERS VALENTIN OVSIENKO AND SERGE TABACHNIKOV Abstract. We describe pairs (p, n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms. 1. Introduction The Hopf fibrations [10] S0 ? Sn ? RPn, S1 ? S2n+1 ? CPn, S3 ? S4n+3 ? HPn, S7 ? S15 ? S8 provide fibrations of spheres whose fibers are great spheres. Algebraic topology imposes severe restrictions on possible dimensions of the spheres and the fibers, the above list actually contains all the existing cases. The study of such fibrations is motivated, in particular, by the classical Blaschke conjecture of differential geometry, see [4]-[7], [18], [12] and [15] for classification of fibrations of spheres by great spheres up to diffeomorphism. Given a fibration of Sn by great spheres Sp, the radial projection from the center on an affine hyperplane yields a fibration of Rn by pairwise skew p-planes. Two affine subspaces of an affine space are called skew if they neither intersect nor contain parallel directions. For example, the projection of the Hopf fibration S1 ? S3 ? S2 gives a fibration of R3 by pairwise skew straight lines (that lie on a nested family of hyperboloids of one sheet), see Figure 1.

• than property


• skew


• hopf


• s3 ?


• problem has


• such fibrations local


• vector fields


• linearly independent


• skew fibrations


• fibration

Sujets

• Eppstein
• Vector field
• Linear independence
• Fibration

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 23
• Langue: English

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ON FIBRATIONS WITH FLAT FIBERS

VALENTIN OVSIENKO AND SERGE TABACHNIKOV

Abstract.We describe pairs (p, n) such thatn-dimensional affine space is fibered by pairwise skewpThe problem is closely related with the theorem of Adams-dimensional affine subspaces. on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.

1.Introduction The Hopf fibrations [10] 0n n1 2n+1n3 4n+3n7 15 8 S→S→RP, S→S→CP, S→S→HP, S→S→S provide fibrations of spheres whose fibers aregreat spherestopology imposes severe. Algebraic restrictions on possible dimensions of the spheres and the fibers, the above list actually contains all the existing cases. The study of such fibrations is motivated, in particular, by the classical Blaschke conjecture of differential geometry, see [4]-[7], [18], [12] and [15] for classification of fibrations of spheres by great spheres up to diffeomorphism. n p Given a fibration ofSby great spheresS, the radial projection from the center on an affine n hyperplane yields a fibration ofRbypairwise skewpaffine subspaces of an affine-planes. Two space are called skew if they neither intersect nor contain parallel directions. For example, the 1 3 2 3 projection of the Hopf fibrationS→S→Sgives a fibration ofRby pairwise skew straight lines (that lie on a nested family of hyperboloids of one sheet), see Figure 1.

Figure 1.The figure is due to David Eppstein, see the Wikipedia article “Skew lines”.

In this paper, we study fibrations p n q R→R→R whose fibers are pairwise skew affine subspaces. We refer to such fibrations as (p, n)-fibrations. The topological restrictions are less prohibitive in this situation, and the list is considerably