The exponential map near conjugate points in 2D hydrodynamics (Q500659)

Let \(M^n\) be a compact Riemannian manifold. Arnold has shown that motions of the ideal incompressible fluid in \(M\) are geodesics on the group \(\mathcal{D}_\mu^s = \mathcal{D}_\mu^s(M)\) of Sobolev class volume-preserving diffeomorphisms. The right-invariant metric on \(\mathcal{D}_\mu^s\) is given at the identity diffeomorphism \(e\) by the \(L^2\) inner product \(\langle v,w \rangle_{L^2}=\int_M \langle v(x),w(x) \rangle d\mu\), where \(v, w \in T_e \mathcal{D}_\mu^s\) are divergence free Sobolev \(H^s\) vector fields on \(M\). This metric is only weak-Riemannian in that the tangent spaces to \(\mathcal{D}_\mu^s\) with the induced inner products are (incomplete) pre-Hilbert spaces. It is known [\textit{D. G. Ebin} et al., Geom. Funct. Anal. 16, No. 4, 850--868 (2006; Zbl 1105.35070)] that the weak-Riemannian exponential map in \(2D\) hydrodynamics is a nonlinear Fredholm map. In this paper the author proves the following result. Let \(M\) be a smooth closed Riemannian manifold of dimension \(2\) and assume \(s>2\). Consider a geodesic \(\eta(t)\) in \(\mathcal{D}_\mu^s\) of the \(L^2\) metric starting from the identity \(e\) with velocity \(v_0\) and let \(\eta(t_c)\) be the first conjugate point to \(e\). Then the weak-Riemannian \(L^2\) exponential map is not injective at \(t_cv_0\).