Title: Conformal Geometry in Lovelock Gravity Theories
Abstract: Lovelock gravity extends Einstein's gravitational theory by incorporating a nonlinear dependency of the curvature terms. Its field equation is the Euler-Lagrange equation derived from the generalized Einstein-Hilbert action. Critical points of this action yield Lovelock metrics, characterizing significant solutions in the theory.
In the context of conformal geometry, Lovelock metrics display similarities to Einstein metrics, elucidating the geometric properties and symmetries inherent in Lovelock gravity theories. What is particularly intriguing is the AdS/CFT correspondence. This correspondence offers a tool for understanding the holographic duality between gravity theories in the bulk (a Riemannian manifold) and conformal field theories on the conformal boundary. This facilitates the study of asymptotic behavior near the conformal boundary.
Analogous to conformally compact Einstein metrics, conformally compact Lovelock metrics are not always smooth near the boundary, which prompts the investigation into boundary obstructions to smoothness and the concept of polyhomogeneity in metric expansion.