Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds

Abstract

Abstract. Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

Universal Positive Mass Theorems

“…It is known that there are ⌊ N 2 ⌋ linearly independent such formulas [5], and finding them is a purely algebraic problem (although somehow tricky), which reduces to finding coefficients (a j ) 1 j N such that the principal symbol of the operator N j=1 a j (P j ) * P j has vanishing second-order part. They can be explicitly obtained through VanderMonde systems [13] or recursive formulas [24].…”

“…The case when N = 2. From [7,13], the number N of irreducible summands in R n ⊗ V is N = 2, i.e. R n ⊗ V = W 1 ⊕ W 2 , in the followng two cases:…”

Comparative Analysis of Spectral Properties of the Hodge–De Rham and Tachibana Operators

“…The operator D * D without the normalizing factor 1 r(r + 1) also appeared in [17,32], but it was proved in [37] that this factor must be present. In [32], the operator D * D was presented without construction and it was mentioned that the operator is elliptic and its kernel consists of conformal Killing forms.…”

“…The method of construction of the operator D in [32] coincides with the method used by Stepanov in [33,35,36]. In [17], the operator D * D was constructed as the operator D *…”

“…In [17], the operator D 3 was called conformal without clarification of this term. Also, in [17], by using the operator D * 3 D 3 , the following estimate of the first eigenvalue λ r 1 of the Hodge-de Rham Laplacian Δ on a compact manifold (M, g) with positive-definite curvature operatorR bounded from below by a number ρ > 0 was obtained:…”