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The geometric sense of the Sasaki connection. (English) Zbl 1048.81032
Summary: For the Riemannian manifold ${M}^{n}$ two special connections are constructed on the sum of the tangent bundle $T{M}^{n}$ and the trivial one-dimensional bundle. These connections are flat if and only if the space ${M}^{n}$ has a constant sectional curvature $±1$. The geometric explanation of this property is given. This construction gives a coordinate-free many-dimensional generalization of the Sasaki connection [R. Sasaki, Soliton equations and pseudospherical surfaces, Nucl. Phys., B 154, 343-357 (1979)]. It is shown that these connections have a close relation to the imbedding of ${M}^{n}$ into Euclidean or pseudo-Euclidean $\left(n+1\right)$-dimension spaces.
##### MSC:
 81R12 Relations of groups and algebras in quantum theory with integrable systems 35Q58 Other completely integrable PDE (MSC2000) 53C07 Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills) 53C42 Immersions (differential geometry) 53C05 Connections, general theory 35Q53 KdV-like (Korteweg-de Vries) equations