  We prove that if $n$ is even, $(M,g)$ is a compact $n$-dimensional Riemannian
manifold whose Pfaffian form is a positive multiple of the volume form, and
$y\in C^{1,\alpha}(M;\mathbb{R}^{n+1})$ is an isometric immersion with
$n/(n+1)< \alpha\leq 1$, then $y(M)$ is a surface of bounded extrinsic
curvature. This is proved by showing that extrinsic curvature, defined by a
suitable pull-back of the volume form on the $n$-sphere via the Gauss map, is
identical to intrinsic curvature, defined by the Pfaffian form. This latter
fact is stated in form of an integral identity for the Brouwer degree of the
Gauss map, that is classical for $C^2$ functions, but new for $n>2$ in the
present context of low regularity.
