2013/07/27 by Gutt, Jean · 5 citations
#37B30 #53D40 #57R17 #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1307.7239
The Conley-Zehnder index associates an integer to any continuous path of symplectic matrices starting from the identity and ending at a matrix which does not admit 1 as an eigenvalue. We give new ways to compute this index. Robbin and Salamon define a generalization of the Conley-Zehnder index for any continuous path of symplectic matrices; this generalization is half integer valued. It is based on a Maslov-type index that they define for a continuous path of Lagrangians in a symplectic vector space (W,Ω), having chosen a given reference Lagrangian V. Paths of symplectic endomorphisms of (\R2n,Ω0) are viewed as paths of Lagrangians defined by their graphs in (W=\R2n⊕ \R2n,Ω=Ω0⊕ -Ω0) and the reference Lagrangian is the diagonal. Robbin and Salamon give properties of this generalized Conley-Zehnder index and an explicit formula when the path has only regular crossings. We give here an axiomatic characterization of this generalized Conley-Zehnder index. We also give an explicit way to compute it for any continuous path of symplectic matrices.