This lower bound is sharp and achieved on the sphere . In fact on the eigenspace for is three dimensional and spanned by the restriction of the coordinate functions from to . Using spherical coordinates , on the two dimensional sphere, set we see easily from the formula for the spherical Laplacian displayed below that Thus the lower bound in Lichnerowicz's theorem is achieved at least in two dimensions. Conversely it was proved by Morio Obata, that if the n-dimensional compact Riemannian manifold without boundary were such that for the first positive eigenvalue one has, then the manifold is isometric to the n-dimensional sphere , the sphere of radius . Proofs of all these statements may be found in the book by Isaac Chavel. Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like the Kohn Laplacian (after Joseph J. Kohn) on a compact CR manifold. Applications there are to the global embedding of such CR manifolds in Tensor Laplacian The Laplace–Beltrami operator can be written using the trace (or contraction) of the iterated covariant derivative associated with the Levi-Civita connection. The Hessian (tensor) of a function is the symmetric 2-tensor , , where df denotes the (exterior) derivative of a function f. Let Xi be a basis of tangent vector fields (not necessarily induced by a coordinate system). Then the components of Hess f are given by This is easily seen to transform tensorially, since it is linear in each of the arguments Xi, Xj. The Laplace–Beltrami operator is then the trace (or contraction) of the Hessian with respect to the metric: . More precisely, this means , or in terms of the metric In abstract indices, the operator is often written provided it is understood implicitly that this trace is in fact the trace of the Hessian tensor. Because the covariant derivative extends canonically to arbitrary tensors, the Laplace–Beltrami operator defined on a tensor T by is well-defined. Laplace–de Rham operator More generally, one can define a Laplacian differential operator on sections of the bundle of differential forms on a pseudo-Riemannian manifold. On a Riemannian manifold it is an elliptic operator, while on a Lorentzian manifold it is hyperbolic. The Laplace–de Rham operator is defined by where d is the exterior derivative or differential and δ is the codifferential, acting as on k-forms, where ∗ is the Hodge star. The