Abstract
AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.
Highlights
The minimal Maslov number of a Lagrangian submanifold in a symplectic manifold is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds, especially monotone Lagrangian submanifolds ([9])
An R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space
In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula
Summary
The minimal Maslov number of a Lagrangian submanifold in a symplectic manifold is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds, especially monotone Lagrangian submanifolds ([9]). R-spaces constitute a nice class of compact embedded totally geodesic Lagrangian submanifolds of Kähler manifolds. Any R-space can be canonically embedded in an Einstein-Kähler C-space and it is a compact embedded monotone Lagrangian submanifold. Oh has worked on the Floer homology of (CPn; RPn) ([10]) and that of real forms of Hermitian symmetric spaces of compact type ([11]), which are nothing but canonically embedded symmetric R-spaces. The intersection theory and Floer homology for two real forms of Hermitian symmetric spaces of compact type are intensively studied by [22], [6], [19], [20], [21], and more recently its generalization to general R-spaces is discussed in [7], [5]
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