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Goertsches, Generators for rational loop groups and geometric applications, arXiv:0803.0029v1 [math.DG
"... Abstract. Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n, C) that satisfy the U(n)reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down a natural set of simpl ..."
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Abstract. Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n, C) that satisfy the U(n)reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write down a natural set of simple elements. Using these simple elements we prove generator theorems for the fundamental representations of the remaining neoclassical groups and most of their symmetric spaces. In order to apply our theorems to submanifold geometry we also obtain explicit dressing and permutability formulae. We introduce a new submanifold geometry associated to G2/SO(4) to which our theory applies. Contents
RESULTS RELATED TO GENERALIZATIONS OF HILBERT’S NONIMMERSIBILITY THEOREM FOR THE HYPERBOLIC PLANE
, 710
"... Abstract. We discuss generalizations of the wellknown theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanif ..."
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Abstract. We discuss generalizations of the wellknown theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As such, we conclude that the only other (noncompact) cases to which this theorem could generalize are the problem of isometric immersions with flat normal bundle of the hyperbolic space H n into a Euclidean space E n+k, n≥2, and the problem of Lagrangian isometric immersions of H n into C n, n ≥ 2. Moreover, there are natural compact counterparts to these problems, and for the compact cases we prove that the theorem does in fact generalize: local embeddings exist, but complete immersions do not. 1.
GRASSMANN GEOMETRIES AND INTEGRABLE SYSTEMS
, 804
"... Abstract. We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann subma ..."
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Abstract. We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given from the recent article [2]. 1.