  For the Riemannian manifold $M^{n}$ two special connections on the sum of the
tangent bundle $TM^{n}$ and the trivial one-dimensional bundle are constructed.
These connections are flat if and only if the space $M^{n}$ has a constant
sectional curvature $\pm 1$. The geometric explanation of this property is
given. This construction gives a coordinate free many-dimensional
generalization of the connection from the paper: R. Sasaki 1979 Soliton
equations and pseudospherical surfaces, Nuclear Phys., {\bf 154 B}, pp.
343-357. It is shown that these connections are in close relation with the
imbedding of $M^{n}$ into Euclidean or pseudoeuclidean $(n+1)$-dimension
spaces.
