to $R$ which allow to state that any $f:R\rightarrow R$ has all derivatives. The main propositions of classical analysis, for example, the properties of derivatives and the Taylor’s formula, are valid for these functions.

In SDG the original theories of tangent bundles, differential forms, connections are constructed. For example, a tangent vector to any object $M$ is a map $t:D\rightarrow M$ and so the tangent bundle of $M$ is an object $M^D$.

The SDG has several models, and particular, so called “well adapted models”, which allow to compare the classical differential geometry with a synthetic one. These models lie in such categories (toposes) $\E$ that there exist a functors which inset the category ${\bf M}$ of $C^\infty$-manifolds in $\E$ (see [@Ko]). For example, we can take a smooth topos as the model for SDG.

In our paper we shall define the basic metrical notions in a context of SDG, and shall show that it is possible to develop pseudo-Riemannian geometry (see [@Gr]) and to write the Einstein’s equations. To this end we shall assume that $R$ satisfies to some properties, which are valid in well adapted models.

We denote as $InvR=\{x\in R\ | \ \exists \ \ y\in R\ \ 