Given a covariant riemannian metric of certain sobolev class (i.e. with square-integrable weak derivatives up to a large enough integer order) on a (compact, if necessary) finite-dimensional smooth manifold, I would like to know the answers to the following related questions:

(1) does its inverse (i.e. the contra-variant type) belong to the same sobolev class?

(2) If a sequence of (covariant) metrics converges in the given sobolev norm to a limit metric, does it follow that the sequence of corresponding inverse metrics converges in the same sobolev norm to the inverse of the above limit metric?

***One of the things that I am iffy about: the inverse of a non-degenerate matrix involves the reciprocal of its determinant. I can believe that the determinant of a metric is a nowhere vanishing sobolev function but I am not sure whether it holds true for its reciprocal.

Thanks very much!

Partial Differential Equations; the statement that $W^{k,p}$ is an algebra is relegated to an exercise at the end of the chapter. $\endgroup$ – Willie Wong Aug 14 '18 at 20:31