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The symplectic area of a geodesic triangle in a Hermitian symmetric space of compact type. (English) Zbl 1509.32006

Summary: Let \(M\) be an irreducible Hermitian symmetric space of compact type, and let \(\omega\) be its Kähler 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.

MSC:

32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
53C22 Geodesics in global differential geometry
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