Curvature of the classical diffeomorphism groups
type: Item
instance of: scholarly article
title: Curvature of the classical diffeomorphism groups
zbMATH Open document ID: 0833.53027
DOI: 10.1007/BF02104943
published in: Siberian Mathematical Journal
publication date: 1995-09-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 794624
zbMATH Keywords: diffeomorphism groups
zbMATH Keywords: contact vector field
zbMATH Keywords: Hamiltonian vector field
zbMATH Keywords: sectional curvature
author: Nicolai K. Smolentsev
MaRDI profile type: Publication
cites work: Q3825006
cites work: Q5719719
cites work: Q4711784
cites work: Reduction of nonholonomic variation problems to isoperimetric ones and connections in principal bundles
cites work: On a constrained variational problem and the space of horizontal paths
full work available at URL: https://doi.org/10.1007/bf02104943
OpenAlex ID: W2046838141
description: scientific article; zbMATH DE number 794624