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Zbl 1048.81032
Shchepetilov, Alexey V.
The geometric sense of the Sasaki connection. (English)
[J] J. Phys. A, Math. Gen. 36, No. 13, 3893-3898 (2003). ISSN 0305-4470

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.

MSC 2000:: *81R12 Relations with integrable systems
35Q58 Other completely integrable PDE
53C07 Special connections and metrics on vector bundles
53C42 Immersions (differential geometry)
53C05 Connections, general theory
35Q53 KdV-like equations

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Zentralblatt MATH Berlin [Germany]