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 $\mathfrak{g}=\mathrm{L}\mathrm{i}\mathrm{e}G$ , we define natural numbers ${\rho }_{G,\mathfrak{s}}(M,\sigma )$ and ${\rho }_{G,\mathfrak{s}}^{\pm }(M,\sigma ,\mathrm{\nabla })$ for a $G$-manifold $(M,\sigma )$ and its affine connection $\mathrm{\nabla }$. 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 ${\rho }_{G,\mathfrak{s}}^{-}(M,\sigma ,\mathrm{\nabla })$ coincides with ${\rho }^{(1)}(\mathfrak{g},\iota )$. Furthermore, we show that ${\rho }_{G,\mathfrak{s}}^{+}(M,\sigma ,\mathrm{\nabla })$ is related to Clifford structures on $M$.