Geodesic
Connection on the group of diffeomorphisms as a bundle over the space of functions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 3, pp. 82-84.

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Jacobian determines a bundle with total space consisting of orientation-preserving diffeomorphisms of a (connected) manifold over the space of positive functions on this manifold (with integral equal to volume for a compact manifold). It is proved that, for the $n$-sphere with standard metric, there is a unique connection on this bundle that is invariant with respect to all isometries of the sphere, and a description of this connection is given.
Keywords: group of diffeomorphisms, manifold of constant curvature, connection. 
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S. M. Gusein-Zade. Connection on the group of diffeomorphisms as a bundle over the space of functions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 3, pp. 82-84. http://geodesic.mathdoc.fr/item/FAA_2021_55_3_a6/

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