  We discuss for which smooth global embeddings of a metric into a Minkowskian
spacetime the Hawking into Unruh mapping takes place. There is a series of
examples of global embeddings into the Minkowskian spacetime (GEMS) with such
mapping for physically interesting metrics. These examples use Fronsdal-type
embeddings for which timelines are hyperbolas. In the present work we show that
for some new embeddings (non Fronsdal-type) of the Schwarzschild and
Reissner-Nordstrom metrics there is no mapping. We give also the examples of
hyperbolic and non hyperbolic type embeddings for the de Sitter metric for
which there is no mapping. For the Minkowski metric where there is no Hawking
radiation we consider a non trivial embedding with hyperbolic timelines, hence
in the ambient space the Unruh effect takes place, and it follows that there is
no mapping too. The considered examples show that the meaning of the Hawking
into Unruh mapping for global embeddings remains still insufficiently clear and
requires further investigations.
