The history of study of Riemannian manifolds in terms of a flat ambient
space began from the Janet-Cartan theorem, which ensures that any N-
dimensional manifold can always be locally (and sometimes global) embedded
in a flat space of dimension N(N+1)/2. Additional symmetries of manifold can
reduce the number of dimensions of the ambient space required for embedding.
For spherically symmetric static manifolds this number is 6. According to
Birkhoff's theorem, the only vacuum solution of Einstein's equations with such symmetry is the Schwarzschild solution.
As Janet-Cartan theorem does not guarantee the uniqueness of embedding,
there are several embedding of the Schwarzschild metric in the 6-dimensional
space. The first of them was founded by Kasner in 1921, but the ambient space
of this embedding has two time-like directions, and so this embedding is not
very interesting for physics. The most famous of the embeddings was offered by Fronsdal in 1959, it is a global embedding in (1+5)-dimensional Minkowskian space. However, a significant drawback of this embedding is that it is not asymptotically flat, i. e. does not turn into the 4-dimensional plane whereas we go far from the gravitating mass. None of several other well-known embeddings of the Schwarzschild metric also has this property. Thus, there is a problem: the insertion of an infinitesimal mass in flat space changes the asymptotic behavior of its embedding, where this mass doesn’t change the physics.
This work presents a global asymptotically flat embedding of the
Schwarzschild solution in (1+5)-dimensional flat space, and solves the problem of the asymptotics in embedding of gravitating mass. The next step would be a generalization of this result to the Regge-Teitelboim approach of gravity.