In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Riemannian symmetric spaces. In this way a classification of the totally geodesic submanifolds in the complex quadric $Q^m := \SO(m+2)/(\SO(2) \times \SO(m))$ is obtained. It turns out that the earlier classification of totally geodesic submanifolds of $Q^m$ by Chen and Nagano is incomplete: in particular a type of submanifolds which are isometric to 2-spheres of radius $\tfrac{1}{2}\sqrt{10}$, and which are neither complex nor totally real in $Q^m$, is missing.Comment: 22 page

Abstract. In this article, the totally geodesic submanifolds in the complex 2-Grassmannian 2 (ℂ +2 ) and in the quaternionic 2-Grassmannian 2 (ℍ +2 ) are classified. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank 2 published by Chen and Nagano (1978) is incomplete. For example, 2 (ℍ +2 ) with ≥ 5 contains totally geodesic submanifolds isometric to a ℍ 2 , its metric scaled such that the minimal sectional curvature is 1 5; they are maximal in 2 (ℍ 7 ). 2 (ℂ +2 ) with ≥ 4 contains totally geodesic submanifolds which are isometric to a ℂ 2 contained in the ℍ 2 mentioned above; they are maximal in 2 (ℂ 6 ). Neither submanifolds are mentioned by Chen and Nagano.

In the first part of this expository article, the most important constructions and classification results concerning totally geodesic submanifolds in Riemannian symmetric spaces are summarized. In the second part, I describe the results of my classification of the totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2.

In this work a spectral theory for 2-dimensional, periodic, complexvalued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally a Jacobi variety and Abel map for the spectral curve is constructed; they are used to describe the 1 2 S. KLEIN Part 6. The Jacobi variety of the spectral curve 153 16. Estimate of certain integrals 153 17. Asymptotic behavior of 1-forms on the spectral curve 168 18. Construction of the Jacobi variety for the spectral curve 183 19. The Jacobi variety and translations of the potential 205 20. Asymptotics of spectral data for potentials on a horizontal strip 219 Part 7. Perspectives 222 21. Perspectives 222 Part 8. Appendices 223 Appendix A. Some infinite sums and products 223 Appendix B. Index of Notations 231 References 235 Part 1. Introduction c has a pole at (λ * , µ * ) , in fact ord Σ (λ * ,µ * ) (µ − d) = 1 is the only possibility. Thus the pole order m of µ−d c equals 2 − 1 = 1 , and hence ord C λ * (c) = m holds.The preceding Proposition shows in particular that if the support of D is contained in the set of regular points of Σ , then we haveHowever, in general it is possible for poles of µ−d c to lie in singularities of Σ , and in these points, the spectral divisor D as defined above does not contain enough information to completely characterize the behavior of the monodromy M(λ) . To handle this case, we need to generalize the concept of a divisor in an appropriate way such that the necessary additional information at the singularities of Σ at which µ−d c is not holomorphic is included. It turns out that the most suitable generalization of the concept of a divisor for the present problem has been introduced by Hartshorne in [Ha2], §1. We now describe Hartshorne's concept of a generalized divisor, which he introduced for general Gorenstein curves in [Ha2] (and later, in even more generality), in our setting, i.e. on a hyperelliptic complex plane curve Σ .As before, we denote by O resp. by M the sheaf of holomorphic functions resp. of meromorphic functions on Σ . A generalized divisor is a subsheaf D of M that is finitely generated over O , and we say that D is positive if O ⊂ D holds. For a positive generalized divisor D , the support of D is the set of points (λ, µ) ∈ Σ for which D (λ,µ) = O (λ,µ) holds. The map Σ → Z Z, (λ, µ) → dim D (λ,µ) /O (λ,µ) defines a divisor on Σ in the usual sense, which we call the underlying classical divisor of D . 20 S. KLEIN Definition 3.2. The (generalized) spectral divisor of the monodromy M(λ) = a(λ) b(λ) c(λ) d(λ) is the subsheaf D of M on Σ generated by the meromorphic functions 1 and µ−d c over O . The generalized spectral divisor D is the proper replacement for the pole divisor of µ−d c in the classical se...