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The Foundations Of Differential Geometry

Basic Algebraic Geometry

On Seifert-manifolds

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

We show that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature. Under the assumption that every geodesic path may be extended to infinity we provide explicit estimates of the growth rate and isoperimetric constant of distance balls in negatively curved tessellations. We show that the assumption about geodesics holds for all tessellations with at least p faces meeting in each vertex and at least q edges bounding each face, where (p,q) ? { (3,6), (4,4), (6,3) } .

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Curvature and Geometry of Tessellating Plane Graphs

Geodesics in non-positively curved plane tessellations

We study the classification problem for left-symmetric algebras with commutation Lie algebra gr(n) in characteristic 0. The problem is equivalent to the classification of etale affine representations of g((n). Algebraic invariant theory is used to characterize those modules for the algebraic group SL(n) which belong to affine etale representations of gr(n). From the classification of these modules we obtain the solution of the classification problem for gl(n). As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.

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Left-symmetric algebras for $\mathfrak{gl}(n)$

We prove that any simply connected special Kaihler manifold admits a canonical immersion as a parabolic affine hypersphere. As an application, we associate a parabolic affine hypersphere to any nondegenerate holomorphic function. We also show that a classical result of Calabi and Pogorelov on parabolic spheres implies Lu's theorem on complete special Kaihler manifolds with a positive definite metric.

/pdf/realisation-of-special-kahler-manifolds-as-parabolic-spheres-55pas6vi0w.pdf

Oliver Baues

Papers

Infra-solvmanifolds and rigidity of subgroups in solvable linear algebraic groups

Curvature and Geometry of Tessellating Plane Graphs

Geodesics in non-positively curved plane tessellations

Left-symmetric algebras for $\mathfrak{gl}(n)$

Realisation of special Kahler manifolds as parabolic spheres