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For every $g\geq 2$ we distinguish real period matrices of real Riemannsurfaces of topological type $(g,0,0)$ from the ones of topological type$(g,k,1)$, with $k$ equal to one or two for $g$ even or odd respectively(Theorem B). To that purpose, we exhibit new invariants of real principallypolarized abelian varieties of orthosymmetric type (Theorem A.1). As a directapplication, we obtain an exhaustive criterion to decide about the existence ofreal points on a real Riemann surface, requiring only a real period matrix ofits and the evaluation of the sign of at most one (real) theta constant(Theorem C). A part of our real, algebro-geometric instruments first appearedin the framework of nonlinear integrable partial differential equations.

Classification of real Riemann surfaces and their Jacobians in the critical case