Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors
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2025
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| author | Gajer, Pawel Ravel, Jacques |
| author_facet | Gajer, Pawel Ravel, Jacques |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10306 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors Gajer, Pawel Ravel, Jacques Differential Geometry Analysis of PDEs Spectral Theory 53C20 (Primary), 58J50, 53C65, 53C21, 35P15 53C20 (Primary), 58J50, 53C65, 53C21, 35P15 (Secondary) 53C20 (Primary), 58J50, 53C65, 53C21, 35P15 (Secondary) 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. |
| title | Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors |
| topic | Differential Geometry Analysis of PDEs Spectral Theory 53C20 (Primary), 58J50, 53C65, 53C21, 35P15 53C20 (Primary), 58J50, 53C65, 53C21, 35P15 (Secondary) 53C20 (Primary), 58J50, 53C65, 53C21, 35P15 (Secondary) |
| url | https://arxiv.org/abs/2508.10306 |