Riemannian exponential maps of the diffeomorphism group of \(\mathbb T^2\)
The authors study the exponential maps on the Lie group of smooth orientation preserving diffeomorphisms of the two-dimensional torus. Let \({\mathfrak D}_+:= {\mathfrak D}_+(\mathbb{T}^2)\) be the group of the orientation preserving \(C^\infty\)-diffeomorphisms of the two-dimensional torus \(\mathbb{T}^2= \mathbb{R}^2/\mathbb{Z}^2\). Then we take the space \(C^\infty(\mathbb{T}^2, \mathbb{R}^2)\) of the smooth vector fields on \(\mathbb{T}^2\) as the Lie algebra \(T_{\text{id}}{\mathfrak D}_+\) of \({\mathfrak D}_+\). Given a nonnegative integer \(k\), define the scalar product \(\langle\cdot, \cdot\rangle_k\) on \(C^\infty(\mathbb{T}^2, \mathbb{R}^2)\) such that \[ \langle u,v\rangle_k:= \sum_{k\geq j\geq 0} \int_{\mathbb{T}^2}\langle(- \Delta)^j u,\;v\rangle dx, \] using a natural scalar product on \(\mathbb{R}^2\). It induces a \(C^1_F\)-smooth weak right invariant Riemannian metric \(\nu^{(k)}\) on \({\mathfrak D}_+\), \[ \nu^{(k)}_\varphi(\xi, \eta):= \langle(dR_\varphi)^{-1}\xi,\;(dR_\varphi)^{-1}\eta\rangle_k \] for \(\forall\varphi\in\mathbb{D}_+\) and \(\forall \xi,\eta\in T_\varphi{\mathfrak D}_+\), where \(R_\varphi\) denotes the right translation by \(\varphi\) on \({\mathfrak D}_+\). Next, the authors go to the definition of geodesic. Namely, for any given positive \(T\), a \(C^2_F\)-smooth curve \(\varphi: [0, T]\to{\mathfrak D}_+\), is said a \(\nu^{(k)}\)-geodesic, if it is a critical point of the functional of \(\gamma\): \[ {\mathcal E}^T_k(\gamma(s, \cdot)):= {1\over 2} \int^T_0 \nu^{(k)}_{\gamma(s,t)}(\dot\gamma(s, t), \dot\gamma(s, t))\,dt, \] where \(\gamma\) runs through all \(C^1_F\)-smooth \({\mathfrak D}_+\)-valued maps defined on \((-\varepsilon,\varepsilon)\times[0, T]\) and satisfying \(\gamma(0,t)\equiv \varphi(t)\), \(\gamma(s, 0)= \varphi(0)\), and \(\gamma(s, T)= \varphi(T)\). Then, first they show that for any positive integer \(k\), there exists an open neighbourhood \(V^{(k)}\) of \(0\) in \(C^\infty(\mathbb{T}^2, \mathbb{R}^2)\) such that for any \(v_0\in V^{(k)}\), there exist unique \(\nu^{(k)}\)-geodesic \(\varphi\) defined on \([-2,2]\) issuing from the identity in \({\mathfrak D}_+\) in the direction vo which depends \(C^1_F\)-smoothly on the initial data. So the Riemanian exponential map, \(\text{Exp}_k|_{V^{(k)}}: v_0\to\varphi(1, v_0)\) is now defined. Secondly, they prove that an open neighbourhood \(\widetilde V^{(k)}\subseteq V^{(k)}\) of \(0\) in \(C^\infty(\mathbb{T}^2,\mathbb{R}^2)\) and an open neighbourhood \(U^{(k)}\) of id in \({\mathfrak D}_+\) exist such that \(\text{Exp}_k|_{\widetilde V^{(k)}}:\widetilde V^{(k)}\to U^{(k)}\) is a \(C^1_F\)-diffeomorphism. However, in the case \(k= 0\), even if one has an exponential map \(\text{Exp}_0\), there is no neighbourhood so that it is a diffeomorphism.
scientific article; zbMATH DE number 5519477