The geometric sense of R. Sasaki connection

Details The geometric sense of R. Sasaki connection The geometric sense of R. Sasaki connection

2002-11-21

arxiv.org/abs/math/0211339v3

J. Phys. A: Math. Gen. V. 36 (2003), pp.3893-3898

Mathematics

Differential Geometry

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

Scientific paper

10.1088/0305-4470/36/13/318

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

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