Asymptotic curvature of \(\Theta\)-metrics
A \(\Theta\)-structure on a smooth compact manifold with boundary \(X\) is a projective class of one-forms \(\Theta\in C^{\infty}(\partial X, T^{\ast}X)\) which is non-vanishing when pulled back to \(\partial X\). Associated to the \(\Theta\)-structure there exists a Lie algebra of vector fields \({\mathcal V}_{\Theta}\), which lifts to the space of all smooth sections of a vector bundle \(^{\Theta}TX\). In the paper under review, given a \(\Theta\)-structure it is defined what is a \(\Theta\)-metric and it is studied the geometry of \(\Theta\)-metrics. The main result of the paper is the expression of the curvature tensor and the Ricci tensor of a \(\Theta\)-metric at boundary points, in terms of the metric tensor and invariants of the \(\Theta\)-structure. As a consequence it is obtained a relation of these invariants that ensures that the curvature tensor is Einstein at the boundary. Moreover, the author determines the structure of the curvature tensor of the induced metric on level sets of a defining function, in terms of the above mentioned invariants. Finally, some examples and applications are given: strictly pseudoconvex domains, Bergman metric, manifolds with contact boundary.
scientific article; zbMATH DE number 1038401