The symplectic area of a geodesic triangle in a Hermitian symmetric space of compact type

TL;DR

This paper investigates the symplectic area of geodesic triangles in compact Hermitian symmetric spaces, providing explicit formulas and conditions for defining these areas unambiguously, extending known results from non-compact cases.

Contribution

It derives an explicit formula for the symplectic area of geodesic triangles in compact Hermitian symmetric spaces, generalizing previous non-compact space results.

Findings

Explicit formula for symplectic area of geodesic triangles in compact spaces

Conditions for unambiguous definition of geodesic triangles

Extension of non-compact space formulas to compact Hermitian symmetric spaces

Abstract

Let $M$ be an irreducible Hermitian symmetric space of compact type, and let $\omega$ be its K\"ahler form. For a triplet $(p_1,p_2,p_3)$ of points in $M$ we study conditions under which a geodesic triangle $\mathcal T(p_1,p_2,p_3)$ with vertices $p_1,p_2,p_3$ can be unambiguously defined. We consider the integral $A(p_1,p_2,p_3)=\int_\Sigma \omega$, where $\Sigma$ is a surface filling the triangle $\mathcal T(p_1,p_2,p_3)$ and discuss the dependence of $A(p_1,p_2,p_3)$ on the surface $\Sigma$. Under mild conditions on the three points, we prove an explicit formula for $A(p_1,p_2,p_3)$ analogous to the known formula for the symplectic area of a geodesic triangle in a non-compact Hermitian symmetric space.