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 $\iota$, and introduced two natural numbers ${\rho }^{(1)}(\mathfrak{g},\iota )$ and ${\rho }^{(2)}(\mathfrak{g},\iota )$. 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 ${\rho }_{G,\mathfrak{s}}(M,\sigma )$ and ${\rho }_{G,\mathfrak{s}}^{\pm }(M,\sigma ,\nabla )$ for a $G$-manifold $(M,\sigma )$ equipped with an affine connection $\nabla$. These are defined in terms of fundamental vector fields on $M$. In a special case, we show that ${\rho }_{G,\mathfrak{s}}(M,\sigma )$ coincides with ${\rho }^{(2)}(\mathfrak{g},\iota )$, and that ${\rho }_{G,\mathfrak{s}}^{-}(M,\sigma ,\nabla )$ coincides with ${\rho }^{(1)}(\mathfrak{g},\iota )$. Furthermore, we show that ${\rho }_{G,\mathfrak{s}}^{+}(M,\sigma ,\nabla )$ is related to Clifford structures on $M$.

18 pages