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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 $TM^n$ and the trivial one-dimensional bundle. 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 Sasaki connection [{\it 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 $(n + 1)$-dimension spaces.

81R12Relations of groups and algebras in quantum theory with integrable systems
35Q58Other completely integrable PDE (MSC2000)
53C07Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills)
53C42Immersions (differential geometry)
53C05Connections, general theory
35Q53KdV-like (Korteweg-de Vries) equations
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