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

TL;DR

This paper provides a geometric interpretation of generalized Hurwitz--Radon numbers via Lie group actions and fundamental vector fields, linking algebraic invariants to geometric structures on manifolds.

Contribution

It introduces new geometric invariants associated with Lie group actions on manifolds, connecting them to previously defined algebraic Hurwitz--Radon numbers and Clifford structures.

Findings

$ ho_{G,rak{s}}(M,\sigma)$ coincides with $ ho^{(2)}(rak g, ext{iota})$ in a special case.

$ ho^{-}_{G,rak{s}}(M,\sigma, abla)$ coincides with $ ho^{(1)}(rak g, ext{iota})$ in a special case.

$ ho^{+}_{G,rak{s}}(M,\sigma, abla)$ relates to Clifford structures on $M$.

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 $\rho^{\pm}_{G,\mathfrak{s}}(M,\sigma,\nabla)$ 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 $M$.