ℤ_p-torus actions on positively curved manifolds

2025/10/30 by Abdullah, Muhammad, Searle, Catherine
#20K01 #53C21 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.26853

Abstract

In this article, we consider closed, positively curved n-manifolds admitting an isometric and effective ℤ_p^r-action with a fixed point, with p a prime, classifying such manifolds up to homotopy equivalence when r satisfies the lower bound given in our main theorem and p ∈ \3,5,7,11,13,17,19,23,29\. In particular, for p=3 and n≥ 1908, we show that r>9n/32. The lower bound we find for this range of primes represents an improvement over the approximately 3n/8 bound found by Fang and Rong in even dimensions and by Ghazawneh in odd dimensions. To obtain this result, we derive a finite-length Elias-Bassalygo bound for q-ary codes that is of independent interest.

Citations

• ℤ₂-actions on positively curved manifolds
• Positive curvature, torus symmetry, and matroids
• Splitting of torus representations and applications in the Grove\n symmetry program
• Nonnegatively and Positively curved manifolds
• Finite isometry groups of 4-manifolds with positive sectional curvature
• Homeomorphism Classification of positively curved manifolds with almost maximal symmetry rank
• Positively curved manifolds with maximal discrete symmetry rank

Related

• ℤ₂-actions on positively curved manifolds
• Positively curved manifolds with maximal discrete symmetry rank
• On Fixed-Point Sets of \Z₂-Tori in Positive Curvature
• Torus actions on rationally elliptic manifolds
• Torus Actions, Fixed-Point Formulas, Elliptic Genera and positive curvature