Generalised Weitzenböck Formulae for Differential Operators in Hörmander Form

Higher Order Derivatives of Heat Semigroups on Spheres and Riemannian Symmetric Spaces

“…There is no problem about integrability, and any choice of parallel translation in T S n T S n will do, [10]. We proceed to calculate this conditional expectation using techniques from [13], see also [8].…”

“…It is simpler to compute the symmetrised versions. For the symmetrised versions the exponential rate is controlled by a Weitzenböck term, in the sense of [13], [8], which for spheres turns out to be essentially the Weitzenböck term for the Lichnerowicz Laplacian, eg see [5]. For general M , the latter has been shown by Bettiol & Mendes, [6], to characterise sectional curvature bounds.…”

“…Our treatment here of S n is as a very special illustrative example of the more general situation described in [14]. We give the necessary geometric background, and give the proof of a simple case concerning the expectation of representations of diffusing Lie group elements.…”

Integration by parts and quasi-invariance for the horizontal Wiener measure on foliated compact manifolds

“…The main reason why we use these connections is that we can not make use of the Bott connection since the adjoint connection to the Bott connection is not metric. We refer to [7,22,30,31] and especially the books [23,24] for a discussion on Weitzenböck-type identities and adjoint connections. Instead we make use of the family of connections first introduced in [2] and only keep the Bott connection as a reference connection.…”

Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds

“…(ii) A Weitzenböck formula in the sub-Riemannian case first appeared in [19,Chapter 2.4]. See also [18]. This formulation assumes that the connection ∇ can be represented as a Le Jan-Watanabe connection.…”