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

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 $\rho^{\pm}_{G,\mathfrak{s}}(M,\sigma,\nabla)$ for a $G$-manifold $(M,\sigma)$ and its 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 $M$.