The theorem of E. Hopf under uniform magnetic fields (Q1277295)

Let \({B}\) denote a closed 2-form on a compact orientable surface \((M,g)\) and \(\Omega:TM\to TM\) a skew-symmetric matrix defined by \(g_x(u,\Omega(\nu))= {B}_x(u,\nu)\) where \(u,\nu\in T_xM\) and \(x\in M\). Physically, \({B}\) has been regarded as a magnetic field whose flow \(\varphi_t: TM\to TM\) is defined by \(\varphi_t(\nu)= \dot c_\nu(t)\), where \(c_\nu\) is a solution curve and \(\dot c\) is the velocity vector. For a geodesic flow, \textit{E. Hopf} [Proc. Natl. Acad. Sci. USA 34, 47-51 (1948; Zbl 0030.07901)] proved that the total curvature of \(M\) without conjugate points is nonpositive, and vanishes if \(M\) is flat. \({B}\) is said to be uniform if \(\nabla{B}=0\), where \(\nabla \) is the metric connection of \(M\). In this paper, the author generalizes the above result of E. Hopf and proves the following Theorem. Let \((M,g)\) be a compact orientable surface with a uniform magnetic field \({B}=b\text{ vol}_M\), \(b\in \mathbb{R}\), where \(\text{vol}_M\) is a canonical volume form of \(M\), and let \(\chi(M)\) denote the Euler characteristic of \(M\). Let \(\exp^{\pm\Omega}: TM\to M\) be the exponential maps associated with \({B}\), with nonsingular values. Then \(-\frac{b^2}{2\pi} \text{vol}(M)\geq \chi(M)\), and equality holds if the curvature \(M\) is \(-b^2\).