Transversally anisotropic product domain

A transversally anisotropic product domain of dimension $n\ge 3$ is a compact connected smooth $n$-dimensional Riemannian domain $(M,g)$ with smooth boundary, together with a compact connected smooth $(n-1)$-dimensional Riemannian manifold with boundary $(M_0,g_0)$ and an isometric embedding of $(M,g)$ as a smooth domain compactly contained in $(\mathbb{R}\times M_0^{\circ },d{x}_1^2\oplus g_0)$. The coordinate $x_1$ on $\mathbb{R}$ is the distinguished Euclidean coordinate; $M_0$ is the transversal manifold. All Laplacians use ${\mathrm{\Delta }}_g={\mathrm{div}}_g{\mathrm{\nabla }}_g$. Neither $M$ nor $M_0$ is assumed orientable. A positive smooth conformal factor is not included in this definition; claims using a conformally related metric must state and prove the required conformal transformation separately.

Riemannian metrics and manifolds

Smooth domains and boundary charts