  The paper is devoted to metric properties of singularities. We investigate
the relations among topology, metric properties and smoothness. In particular,
we present some higher dimensional analogous of Mumford's theorem on smoothness
of normal surfaces. For example, we prove that a complex analytic set, with an
isolated singularity at $0$, is smooth at $0$ if and only if it is locally
metrically conical at $0$ and its link at $0$ is a homotopy sphere.
