A Geometric Interpretation of Generalized Hurwitz--Radon Numbers Defined by Kannaka--Tojo

Abstract

The Hurwitz--Radon number originates in the composition problem of quadratic forms and is related to the maximum number of pointwise linearly independent vector fields on spheres. Kannaka--Tojo [arXiv:2602.04544] reformulated the Hurwitz--Radon number in the setting of a real reductive Lie algebra $\mathfrak g$ and its faithful representation $ι$, and introduced two natural numbers $ρ^{(1)}(\mathfrak g,ι)$ and $ρ^{(2)}(\mathfrak g,ι)$. For classical Lie algebras and their standard representations, these two numbers coincide except for a few cases. In this paper, fixing a Lie group $G$ and a subspace $\mathfrak{s} $ of $ \mathfrak g = \mathrm{Lie}~G$ , we define natural numbers $ρ_{G,\mathfrak{s}}(M,σ)$ and $ρ^{\pm}_{G,\mathfrak{s}}(M,σ,\nabla)$ for a $G$-manifold $(M,σ)$ and its affine connection $\nabla$. These are defined in terms of fundamental vector fields on $M$. In a special case, we show that $ρ_{G,\mathfrak{s}}(M,σ)$ coincides with $ρ^{(2)}(\mathfrak g,ι)$, and that $ρ^{-}_{G,\mathfrak{s}}(M,σ,\nabla)$ coincides with $ρ^{(1)}(\mathfrak g,ι)$. Furthermore, we show that $ρ^{+}_{G,\mathfrak{s}}(M,σ,\nabla)$ is related to Clifford structures on $M$.