Papers by Jost Eschenburg
Matemática Contemporânea
For an equivariant embedding of a compact symmetric space X = G/K into a Euclidean G-space the fo... more For an equivariant embedding of a compact symmetric space X = G/K into a Euclidean G-space the following statements are equivalent: (a) The embedding is extrinsic symmetric. (b) The maximal torus T X of X is rectangular and the representation of G has lowest possible highest weight. (c) The maximal torus T X is embedded as a Clifford torus (an extrinsic product of planar circles).
Hermitian–Grassmannian Submanifolds, 2017
We sketch a geometric proof of the classical theorem of Atiyah, Bott, and Shapiro [3] which relat... more We sketch a geometric proof of the classical theorem of Atiyah, Bott, and Shapiro [3] which relates Clifford modules to vector bundles over spheres. Every module of the Clifford algebra Cl k defines a particular vector bundle over S k+1 , a generalized Hopf bundle, and the theorem asserts that this correspondence between Cl k-modules and stable vector bundles over S k+1 is an isomorphism modulo Cl k+1-modules. We prove this theorem directly, based on explicit deformations as in Milnor's book on Morse theory [8], and without referring to the Bott periodicity theorem as in [3].
Minimal surfaces in euclidean 3-space, i.e. conformal harmonic maps, enjoy two important properti... more Minimal surfaces in euclidean 3-space, i.e. conformal harmonic maps, enjoy two important properties: They allow a cir-cle of isometric deformations rotating the principal curvature direc-tions, the so called associated family, and they are obtained as the real part of holomorphic functions into C 3 . These properties are shared by arbitrary (pluri-)harmonic maps into euclidean n-space. Replacing R n with an arbitrary symmetric space leads to similar results, but the rôle of C n is played by an infinite dimensional com-plex homogeneous space acted on by a twisted loop group. We give a survey of the development of this theory from our view point, and we discuss applications to the construction of (pluri-)harmonic maps into symmetric spaces and their rank restrictions.
Journal of Algebra, 2014
We show that for any prime number n = 2r +1 5 there exist r planar tilings with self-similar vert... more We show that for any prime number n = 2r +1 5 there exist r planar tilings with self-similar vertex set and the symmetry of a regular n-gon (D n-symmetry). The tiles are the rhombi with angle πk/n for k = 1,. .. , r.
Minimal surfaces in euclidean 3-space, i.e. conformal harmonic maps, enjoy two important properti... more Minimal surfaces in euclidean 3-space, i.e. conformal harmonic maps, enjoy two important properties: They allow a circle of isometric deformations rotating the principal curvature directions, the so called associated family, and they are obtained as the real part of holomorphic functions into C. These properties are shared by arbitrary (pluri-)harmonic maps into euclidean n-space. Replacing R with an arbitrary symmetric space leads to similar results, but the rôle of C is played by an infinite dimensional complex homogeneous space acted on by a twisted loop group. We give a survey of the development of this theory from our view point, and we discuss applications to the construction of (pluri-)harmonic maps into symmetric spaces and their rank restrictions.
Mathematische Zeitschrift, 1999
Bulletin of the London Mathematical Society, 2006
Contents 1. Submanifolds with ∇α = 0 are extrinsic symmetric. 1 2. Extrinsic symmetric spaces (ES... more Contents 1. Submanifolds with ∇α = 0 are extrinsic symmetric. 1 2. Extrinsic symmetric spaces (ESS) split extrinsically. 2 3. The indecomposable ESS. 3 4. ESS are certain isotropy orbits of symmetric spaces. 5 5. How to classify ESS. 7 6. ESS are real forms of hermitian symmetric spaces. 8 7. ESS are midpoint components between center elements. 8. Maximal tori of ESS are products of planar circles. 9. Isometries of ESS are extrinsic. 10. ESS have a noncompact transformation group. 11. ESS contain their noncompact duals. 12. ESS in symmetric spaces come from ESS in euclidean space.
Pluriharmonic Maps, Loop Groups and Twistor Theory
ABSTRACT We consider pluriharmonic maps from simply connected complexmanifolds M to compact symme... more ABSTRACT We consider pluriharmonic maps from simply connected complexmanifolds M to compact symmetric spaces G/K. Following [11]we introduce the loop parameter – and thus the associated familyof – in a geometric fashion. We define an extended framingglobally on M, but with values in /K, where denotes a loop group associated with G. Locally we obtain adescription of pluriharmonic maps via normalized potentials similar tothat of Dorfmeister, Petit and Wu. For dimM > 1 thesepotentials satisfy a `curved flat'' condition and they characterizepluriharmonic maps. We also briefly discuss the dressing action on theset of pluriharmonic maps. Finally, as special cases of the generaltheory, we discuss the isotropic case and pluriharmonic maps into Liegroups.
Mathematische Zeitschrift, 1987
Die Gleichung 5. Grades: Ist Mathematik erzählbar?
Mathematische Semesterberichte, 2000
Manuscripta Mathematica, 2013
In the PhD thesis of Huang with Leung (Huang, A uniform description of Riemannian symmetric space... more In the PhD thesis of Huang with Leung (Huang, A uniform description of Riemannian symmetric spaces as Grassmannians using magic square, www.ims.cuhk.edu.hk/ leung/; Huang and Leung, Math Ann 350:76-106, 2010), all compact symmetric spaces are represented as (structured) Grassmannians over the algebra KL := K ⊗ R L where K, L are real division algebras. This was known in some (infinitesimal) sense for exceptional spaces (see Baez, Bull Am Math Soc 39:145-205, 2001); the main purpose in Huang (www.ims. cuhk.edu.hk/~leung/) and Huang and Leung (Math Ann 350:76-106, 2010) was to give a similar description for the classical spaces. In the present paper we give a different approach to this result by investigating the fixed algebras B of involutions on A = KL with halfdimensional eigenspaces together with the automorphism groups of A and B. We also relate the results to the classification of self-reflective submanifolds in Chen and Nagano (Trans
Manuscripta Mathematica, 1992
Biquotients are non-homogeneous quotient spaces of Lie groups. Using the Serre spectral sequence ... more Biquotients are non-homogeneous quotient spaces of Lie groups. Using the Serre spectral sequence and the method of Borel, we compute the cohomology algebra of these spaces in cases where the Lie group cohomology is not too complicated. Among these are the biquotients which are known to carry a metric of positive curvature.
Manuscripta Mathematica, 2011
We extend Ferus' characterization of extrinsic symmetric spaces to ambient spaces with indefinite... more We extend Ferus' characterization of extrinsic symmetric spaces to ambient spaces with indefinite inner products.
Journal of Geometric Analysis, 2000
Geometriae Dedicata, 1989
REDUCTION OF CODIMENSION OF SURFACES Let M be a 2-dimensional smooth manifold, Q an n-dimensional... more REDUCTION OF CODIMENSION OF SURFACES Let M be a 2-dimensional smooth manifold, Q an n-dimensional space form of constant sectional curvature and x: M-, Q an immersion. We say that we can reduce the codimension to k < n-2 if there is a (k + 2)-dimensional totally geodesic submanifold Q' c Q such that x(M) c Q'. We always equip M with the induced metric. Let N be the normal bundle with its induced connection D and H the mean curvature vector of the immersion. Our first theorem generalizes a result of [1] and [6]:
Differential Geometry and its Applications, 2004
We investigate the local geometry of a class of Kähler submanifolds M ⊂ R n which generalize surf... more We investigate the local geometry of a class of Kähler submanifolds M ⊂ R n which generalize surfaces of constant mean curvature. The role of the mean curvature vector is played by the (1, 1)-part (i.e. the dz i dz j-components) of the second fundamental form α, which we call the pluri-mean curvature. We show that these Kähler submanifolds are characterized by the existence of an associated family of isometric submanifolds with rotated second fundamental form. Of particular interest is the isotropic case where this associated family is trivial. We also investigate the properties of the corresponding Gauss map which is pluriharmonic.
Codimension of immersions with parallel pluri-mean curvature
Differential Geometry and its Applications, 2009
ABSTRACT Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant... more ABSTRACT Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.
Differential Geometry and its Applications, 1992
We describe the geometry and the topology of a compact simply connected positively curved Riemann... more We describe the geometry and the topology of a compact simply connected positively curved Riemannian g-manifold F' which is related to the flag manifold F over cP2, and an infinite series of simply connected circle bundles over F', also with positive sectional curvature. All of these spaces are biquotients of the Lie group SU(3) and they are not homeomorphic to a homogeneous space of positive curvature.
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Papers by Jost Eschenburg