ON FIBRATIONS WITH FLAT FIBERS

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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.

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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

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