  We show how Lasry-Lions's result on regularization of functions defined on
$\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of
distances can be extended to (finite or infinite dimensional) Riemannian
manifolds $M$ of bounded sectional curvature. More specifically, among other
things we show that if the sectional curvature $K$ of $M$ satisfies $-K_0\leq
K\leq K_0$ on $M$ for some $K_0>0$, and if the injectivity and convexity radii
of $M$ are strictly positive, then every bounded, uniformly continuous function
$f:M\to\mathbb{R}$ can be uniformly approximated by globally $C^{1,1}$
functions defined by $$ (f_{\lambda})^{\mu}=\sup_{z\in M}\inf_{y\in
M}\{f(y)+\frac{1}{2\lambda} d(z,y)^{2}-\frac{1}{2\mu}d(x,z)^2\} $$ as $\lambda,
\mu\to 0^{+}$, with $0<\mu<\lambda/2$. Our definition of (global) $C^{1,1}$
smoothness is intrinsic and natural, and it reduces to the usual one in flat
spaces, but we warn the reader that, in the noncompact case, this definition
differs from other notions of (rather local) $C^{1,1}$ smoothness that have
been recently used, for instance, by A. Fathi and P. Bernard (based on charts).
The importance of this regularization method lies (rather than on the degree of
smoothness obtained) on the fact that the correspondence $f\mapsto
(f_{\lambda})^{\mu}$ is explicit and preserves many significant geometrical
properties that the given functions $f$ may have, such as invariance by a set
of isometries, infima, sets of minimizers, ordering, local or global
Lipschitzness, and (only when one additionally assumes that $K\leq 0$) local or
global convexity. We also give two examples showing that this result completely
fails, even for (nonflat) Cartan-Hadamard manifolds, whenever $f$ or $K$ are
not bounded.
