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 g and its faithful representation ι, and introduced two natural numbers ρ⁽¹⁾(g,ι) and ρ⁽²⁾(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 = \operatorname{Lie}(G)$ , we define natural numbers ρ_G,s(M,σ) and ρ^±_G,s(M,σ,∇) for a G-manifold (M,σ) equipped with an affine connection ∇. These are defined in terms of fundamental vector fields on M. In a special case, we show that ρ_G,s(M,σ) coincides with ρ⁽²⁾(g,ι), and that ρ⁻_G,s(M,σ,∇) coincides with ρ⁽¹⁾(g,ι). Furthermore, we show that ρ⁺_G,s(M,σ,∇) is related to Clifford structures on M.

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BibTeX RIS

BibTeX

@misc{miyaji2026geometric,
  title = {{A Geometric Interpretation of Generalized Hurwitz--Radon Numbers Defined by Kannaka--Tojo}},
  author = {Muneto Miyaji},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.00704},
  eprint = {2605.00704},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.00704},
}

RIS

TY  - UNPB
TI  - A Geometric Interpretation of Generalized Hurwitz--Radon Numbers Defined by Kannaka--Tojo
AU  - Muneto Miyaji
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2605.00704
UR  - https://arxiv.org/abs/2605.00704
AB  - 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 g and its faithful representation ι, and introduced two natural numbers ρ^(1)(g,ι) and ρ^(2)(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 = \operatorname{Lie}(G)$ , we define natural numbers ρ_G,s(M,σ) and ρ^±_G,s(M,σ,∇) for a G-manifold (M,σ) equipped with an affine connection ∇. These are defined in terms of fundamental vector fields on M. In a special case, we show that ρ_G,s(M,σ) coincides with ρ^(2)(g,ι), and that ρ^−_G,s(M,σ,∇) coincides with ρ^(1)(g,ι). Furthermore, we show that ρ^+_G,s(M,σ,∇) is related to Clifford structures on M.
ER  -

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