Abstract. The nonexistence of isometric embeddings m q → n p with p = q is proved. The only exception is q = 2, p ∈ 2N, in which case an isometric embedding exists if n is sufficiently large, n ≥ N (m, p). Some lower bounds for N (m, p) are obtained by using the equivalence between the isometric embeddings in question and the cubature formulas for polynomial functions on projective spaces. Even though only the quaternion case is new, the exposition treats the real, complex, and quaternion cases simultaneously.
Abstract. The polynomial functions on a projective space over a field K = R, C or H come from the corresponding sphere via the Hopf fibration. The main theorem states that every polynomial function φ(x) of degree d is a linear combination of "elementary" functions | x, · | d .1. Spaces and operators. The classical projective spaces are KP m−1 where K is one of three fields R, C, H and m ≥ 2. The quaternion field H is noncommutative in contrast to R and C. However, R ⊂ H and each real number commutes with all quaternions. In general, K is an associative unital algebra over R of dimension δ = δ(K) = 1, 2, 4 for K = R, C, H respectively. The standard conjugation α → α is an involutive automorphism of K (or anti-automorphism if K = H since αβ = βα in this case). For any α ∈ K the real number Re α = 1 2 (α + α) is the real part of α. (If K = R we set α = α.) The space KP m−1 can be constructed starting with K m that consists of all m-tuples x = (ξ i ), ξ i ∈ K. This is an m-dimensional right (for definiteness) linear space over K; the addition in K m is standard, the multiplication by a scalar α ∈ K is xα = (ξ i α). Moreover, K m is a Euclidean space provided with the inner productThe properties of the latter are standard but with the fixed order of factors in the relations xα, y = α x, y , x, yα = x, y α if K = H. The corresponding Euclidean norm on K m is (2) x = x, x = |ξ i | 2 .
If q = Q, then the matrix ~ satisfies the following evolution system with respect to t [5, 6]:We use this system and the evolution system for r to evaluate the derivative (14). We express the derivative 0t~ (2) via system (15), the derivative 0re (2) via the evolution system for r and the derivativeOtU via system (7). As a result, we get an expression that does not contain derivatives with respect to t.After some transformations, we obtain (8).
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