Local and global algebraic structures in general relativity (Q1824238)

Two important questions are considered. First, to what extent is the Petrov classification and related canonical forms smooth over the manifold if the tensor is? No surprises here; everything is as one might have thought. Second, suppose a smooth, symmetric second-order tensor has the same algebraic type at all points and has all real eigenvalues. Then the first problem shows that these eigenvalues are global smooth functions. But are there global, smooth eigenvectors ? Roughly, yes, if the manifold is simply connected (and the same for the Weyl tensor) so the topology of a manifold supporting certain types of energy tensor/Weyl tensor is restricted.