Curvature of the classical diffeomorphism groups
The author establishes formulas for the sectional curvatures of the classical diffeomorphism groups \(D_\mu\), \(D_\omega\), \(D_\theta\). For an orientable closed smooth Riemannian manifold \(M\) let \(D_\mu\) stand for the group of smooth diffeomorphisms of \(M\) preserving the Riemannian volume element \(\mu\) on \(M\). The sectional curvature of the group \(D_\mu\) in the direction of the two-dimensional piece \(\sigma\) determined by an orthogonal pair of vectors \(X,Y\in T_e D_\mu\) is given by the formula \[ \begin{aligned} K_\sigma= &- {\textstyle {1\over 2}} \bigl(X, \bigl[[X, Y], Y\bigr]\bigr)- {\textstyle {1\over 2}} \bigl( \bigl[X, [X,Y ]\bigr], Y\bigr)-\\ &-{\textstyle {3\over 4}} \bigl( [X,Y ], [X,Y ]\bigr)- \bigl( P_e (\nabla_X X), P_e (\nabla_Y Y) \bigr)+\\ &+{\textstyle {1\over 4}} \bigl( P_e (\nabla_X Y+ \nabla_Y X), P_e (\nabla_X Y+ \nabla_Y X) \bigr), \end{aligned} \] where \(P_e\) is some projection. The author obtains analogous formulas for the group \(D_\omega\) of smooth symplectic diffeomorphisms and for the group \(D_\theta\) of contact diffeomorphisms.
scientific article; zbMATH DE number 794624