Strong approximation of quadrics and representations of quadratic forms

Let V be an indefinite quadratic space over a number field F and U be a nondegenerate subspace of V . Suppose that M is a lattice on V , and that N is a lattice on U which is represented by M locally everywhere. The main result of this paper is a necessary and sufficient condition for which there exists a representation of N by M that approximates a given family of local representations. This is applied to determine when the variety of representations of U by V has strong approximation with respect to a finite set of primes of F that contains all the archimedean primes.