Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors

2025/08/14 by Gajer, Pawel, Ravel, Jacques
#35P15 (Secondary) #35P15 (Secondary) 53C20 (Primary) #35P15 53C20 (Primary) #53C20 (Primary) #53C21 #53C65 #58J50 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.10306

Abstract

We introduce two pointwise subspace averages of sectional curvature on a d-dimensional plane Pi in T_p M: (i) the intrinsic mean Ricci (the average of sectional curvatures of 2-planes contained in Pi); and (ii) the normal (mixed) mean Ricci (the average of sectional curvatures of 2-planes spanned by one vector in Pi and one in Pi^perp). Using Jacobi-field expansions, these means occur as the r²/6 coefficients in the intrinsic (d-1)-sphere and normal (n-d-1)-sphere volume elements. A direct consequence is a Bochner--Weitzenboeck identity for simple d-vectors V (built from an orthonormal frame X₁,...,X_d with Pi = spanX_i): the curvature term equals d(n-d) times the normal mean Ricci of Pi. This yields two immediate applications: (a) a Bochner vanishing criterion for harmonic simple d-vectors under a positive lower bound on the normal mean Ricci; and (b) a Lichnerowicz-type lower bound for the first eigenvalue of the Hodge Laplacian on simple d-eigenfields.

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