  We present results relevant to the relation between quantum effects in a
Riemannian space and on the surface appearing as a result of its isometric
embedding in a flat space of a higher dimension. We discuss the mapping between
the Hawking effect fixed by an observer in the Riemannian space with a horizon
and the Unruh effect related to an accelerated motion of this observer in the
ambient space. We present examples for which this mapping holds and examples
for which there is no mapping. We describe the general form of the hyperbolic
embedding of the metric with a horizon smoothly covering the horizon and prove
that there is a Hawking into Unruh mapping for this embedding. We also discuss
the possibility of relating two-point functions in a Riemannian space and the
ambient space in which it is embedded. We obtain restrictions on the geometric
parameters of the embedding for which such a relation is known.
