The Geodesic Flow Acting on Lagrangian Subspaces
This chapter describes how the geodesic flow acts on Lagrangian subspaces. We introduce Lagrangian subspaces and Lagrangian submanifolds and we show an important property of the vertical subbundle which we call the twist property of the vertical subbundle. This property reflects the fact that the geodesic flow arises from a second order differential equation on TM. Next we derive the Riccati equations, after which we introduce the Grassmannian bundle of Lagrangian subspaces and show how to attach an index, the Maslov index, to every closed curve of Lagrangian subspaces. The definition used of the Maslov index follows Mañé in [Ma1] and it is particularly adapted to the Riccati equations. This allows us to show that the lift of the geodesic flow to the Grassmannian bundle of Lagrangian subspaces is transverse to the Maslov cycle. This important property reflects the convexity of the unit spheres in tangent spaces.
KeywordsRiccati Equation Symplectic Form Lagrangian Submanifolds Geodesic Flow Maslov Index
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