Quadratic forms between euclidean spheres

A generalization of Cartan’s theorem on isoparametric cubics

“…More specifically, let us denote by σ(k), k ≥ 1, the minimal possible value of l for which there exists a nonconstant homogeneous quadratic map S k → S l . Then the theorem of P. Yiu [6,Theorem 4] (see also [7] for general polynomial maps) yields a recursive formula for σ(k):…”

“…Note also that our assumption q ≥ 2 implies by virtue of (18) that the image of y(S p−1 ) in S q−1 is distinct from a point. Then one result of P. Yiu [6] provides an obstruction for a nonconstant quadratic map to exist if the dimension q − 1 of the target sphere is too small. More specifically, let us denote by σ(k), k ≥ 1, the minimal possible value of l for which there exists a nonconstant homogeneous quadratic map S k → S l .…”

Circles and Quadratic Maps Between Spheres

“…Yiu [6] described all pairs of positive integers m, n such that there is a non-constant quadratic map of S m to S n . Namely, n κ(m), where the Yiu function κ is defined recurrently as follows: κ(2 t + m) = 2 t , 0 m < ρ(2 t ) 2 t + κ(m), ρ(2 t ) m < 2 t Let f : S m → S n be any map between unit spheres.…”

“…Results of [5,6] lead to some interesting consequences regarding roundings, including the following:…”

“…There exists a nondegenerate rounding : (R m , 0) → (R n , 0) if and only if n κ(m), where κ is an explicit function introduced in [6].…”