# Quadratic Spherical Maps

• Takashi Ono
Chapter
Part of the The University Series in Mathematics book series (USMA)

## Abstract

Let K be an infinite field of characteristic #2. For vector spaces X, Y over K of finite dimension, we defined a quadratic mapf: X→ Y by the following conditions :
$$f(ax) = {a^2}f(x),a \in K,x \in X$$
(5.1)
$$(x,y) \mapsto \frac{1}{2}[f(x + y) - f(x) - f(y)],x,y \in XS$$
(5.2)
is bilinear [see (1.2) and (1.3)]. We denoted by Q(X, Y) the set of all such maps. In what follows, we assume that X, Y are both nonsingular quadratic spaces with quadratic forms q x , q Y , respectively, and there are x∈X, y∈Y such that q x (x) = qy(y) = 1. Therefore the unit spheres S x , S Y are nonempty:
$${S_X} = \{ x \in X;{q_X}(X) = 1\} ,{S_Y} = \{ y \in Y;{q_Y}(y) = 1\}$$
.

## Keywords

Quadratic Form Orthogonal Basis Homotopy Theory Infinite Field Witt Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

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A. Hattori, Groups and Their Representations, Kyoritsu, Tokyo (1967).Google Scholar
2. 2.
P. J. Hilton, An Introduction to Homotopy Theory, Cambridge Tracts in Math., 43, Cambridge University Press (1953).
3. 3.
H. Hopf, Selecta: Heinz Hopf (Springer, New York-Heidelberg-Berlin, 1964).
4. 4.
N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, N J (1951).
5. 5.
E. L. Steifel and G. Scheifele, Linear and Regular Celestial Mechanics, Die Grund, d. math. Wiss., 174, Springer, New York-Heidelberg-Berlin, (1971).
6. 6.
R. Wood, Inventiones Math. 5, 163 (1968).

## Copyright information

© Takashi Ono 1994

## Authors and Affiliations

• Takashi Ono
• 1
1. 1.The Johns Hopkins UniversityBaltimoreUSA