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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 ±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 (n+1)-dimension spaces.
MSC:
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