We discuss 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 correspondence 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 correspondence holds and examples for which there is no correspondence. 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 correspondence between the Hawking and Unruh effects 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.