The geometric sense of R. Sasaki connection

Abstract

For the Riemannian manifold $M^n$ two special connections on the sum of the tangent bundle $T{M}^n$ 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.

7 pages, the key reference to the paper of Min-Oo is included in the second version

References in corpus (1)

• Scalar second order evolution equations possessing an irreducible sl${}_2$-valued zero curvature representation