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

الأصل · EN

Let M be an irreducible Hermitian symmetric space of compact type, and let ω be its Kähler form. For a triplet (p₁,p₂,p₃) of points in M we study conditions under which a geodesic triangle T(p₁,p₂,p₃) with vertices p₁,p₂,p₃ can be unambiguously defined. We consider the integral A(p₁,p₂,p₃)=∫Σω, where Σ is a surface filling the triangle T(p₁,p₂,p₃) and discuss the dependence of A(p₁,p₂,p₃) on the surface Σ. Under mild conditions on the three points, we prove an explicit formula for A(p₁,p₂,p₃) analogous to the known formula for the symplectic area of a geodesic triangle in a non-compact Hermitian symmetric space.