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arXiv · 2508.10306

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

Pawel Gajer·Jacques Ravel
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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 r^2/6 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 X_1,...,X_d with Pi = span{X_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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Pawel Gajer, Jacques Ravel. 2025-08-14. Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors. https://arxiv.org/abs/2508.10306

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