According to Gromov's *Metric Structures for Riemannian and Non-Riemannian Spaces* every limit $V_0$ of sequences in the class of manifolds $V$ with $|K(V)| \leq 1$ and $\mathrm{InjRad}(V) \geq \rho > 0$ is a Riemannian $C^{1,1}$-manifold, which according to Gromov's *ad hoc* definition satisfies the following (among other things, see p. 387):

i. There exists a $C^{1,1}$-atlas (differentiable w/ Lipschitz derivative), and the metric tensor is Lipschitz in these coordinates (using harmonic coordinates I think one should obtain that $g$ is $C^{1,1}$ as well).

ii. The squared distance function is locally $C^{1,1}$.

iii. There exists a bounded measurable quadratic form $B_S$ (one for each hyperplane in $T_v V_0$) satisfying the following *tube formula* (see p. 372):

For every hypersurface $W\subset V_0$ through $v$ with shape operator $A_0$ and $T_v W = S$, $$ \frac{d}{dt} A^\ast_t \bigg\vert_{t=0} = - (A_0^\ast)^2 + B_S,$$ where $A_t^\ast$ denotes the pullback of the shape operator $A_t$ of the hypersurface $W_t$ which is obtained from $W$ by equidistant translation.

The interesting thing about $B_S$ is that it can be used to define sectional curvature by setting $$ K(\tau_1 \land \tau_2) = - g(B_S(\tau_1),\tau_2),$$ where $\tau_1 \bot S$ and $\tau_2 \in S$.

**Question 1:** How is B_S obtained? How does one see that it is measurable? I'm assuming that using harmonic coordinates one sees that $A$ is Lipschitz and thus differentiable almost everywhere and then obtains $B_S$ in terms of the derivative of $A$ using the tube formula as definition for $B_S$...

**Question 2:** Can one obtain a similar definition for sectional curvature if the metric tensor is only $C^{1,\alpha}$ (differentiable with H\"older continuous derivative)?