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Dive into the research topics where Oliver Goertsches is active.

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Featured researches published by Oliver Goertsches.


Geometriae Dedicata | 2007

On the Geometry of the Orbits of Hermann Actions

Oliver Goertsches; Gudlaugur Thorbergsson

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes of the orbits and a new proof of a Weyl-type integration formula for Hermann actions.


Crelle's Journal | 2016

EQUIVARIANT BASIC COHOMOLOGY OF RIEMANNIAN FOLIATIONS

Oliver Goertsches; Dirk Töben

Abstract The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.


Journal of Topology | 2010

TORUS ACTIONS WHOSE EQUIVARIANT COHOMOLOGY IS COHEN-MACAULAY

Oliver Goertsches; Dirk Töben

We study Cohen–Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang–Skjelbred Lemma, and its stronger version, the exactness of an Atiyah–Bredon sequence, hold. The main difference is that the fixed-point set is replaced by the union of lowest dimensional orbits. We find sufficient conditions for the Cohen–Macaulay property such as the existence of an invariant Morse–Bott function whose critical set is the union of lowest dimensional orbits, or open-face-acyclicity of the orbit space. Specializing to the case of torus manifolds, that is, 2r-dimensional orientable compact manifolds acted on by r-dimensional tori, the latter is similar to a result of Masuda and Panov, and the converse of the result of Bredon that equivariantly formal torus manifolds are open-face-acyclic.


Transformation Groups | 2011

Torsion in equivariant cohomology and Cohen-Macaulay G-actions

Oliver Goertsches; Sönke Rollenske

We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.


Israel Journal of Mathematics | 2017

Localization of Chern–Simons type invariants of Riemannian foliations

Oliver Goertsches; Hiraku Nozawa; Dirk Töben

We prove an Atiyah–Bott–Berline–Vergne type localization formula for Killing foliations in the context of equivariant basic cohomology. As an application, we localize some Chern–Simons type invariants, for example the volume of Sasakian manifolds and secondary characteristic classes of Riemannian foliations, to the union of closed leaves. Various examples are given to illustrate our method.


Journal of The London Mathematical Society-second Series | 2015

On the boundary behavior of left-invariant Hitchin and hypo flows

Florin Belgun; Vicente Cortés; Marco Freibert; Oliver Goertsches

We investigate left-invariant Hitchin and hypo flows on


Transactions of the American Mathematical Society | 2011

Generators for rational loop groups

Neil Donaldson; Daniel Fox; Oliver Goertsches

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Journal of Geometry and Physics | 2018

Locally conformally symplectic convexity

Florin Belgun; Oliver Goertsches; David Petrecca

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Mathematische Annalen | 2012

Equivariant cohomology of K-contact manifolds

Oliver Goertsches; Hiraku Nozawa; Dirk Töben

6


Topology and its Applications | 2014

Equivariant cohomology of cohomogeneity one actions

Oliver Goertsches; Augustin-Liviu Mare

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Hiraku Nozawa

École normale supérieure de Lyon

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Florin Belgun

Humboldt University of Berlin

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Neil Donaldson

University of California

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