arXiv subjects
Publications and source records attributed to Oliver Goertsches.
We show that the isotropy action of a homogeneous space $G/K$, where $G$ and $K$ are compact, connected Lie groups and $K$ is defined by an automorphism on $G$, is equivariantly formal and that $(G, K)$ is a Cartan pair.
Let T be a torus of dimension at least k and M a T-manifold. M is a GKM_k-manifold if the action is equivariantly formal, has only isolated fixed points, and any k weights of the isotropy representation in the fixed points are linearly independent. In this paper we compute the cohomology rings with real and integer coefficients of GKM_3- and GKM_4-manifolds which admit invariant metrics of positive sectional curvature.
In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coKähler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we distinguish between (quasi-)regular and irregular structures; in the regular case, the K-cosymplectic manifold turns out to be a flat circle bundle over an almost Kähler manifold. We investigate de Rham and basic cohomology of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector fields and group actions on such manifolds. The deformations of type I and II in the contact setting have natural analogues for cosymplectic manifolds; those of type I can be used to show that compact K-cosymplectic manifolds always carry quasi-regular structures. We consider Hamiltonian group actions and use the momentum map to study the equivariant cohomology of the canonical torus action on a compact K-cosymplectic manifold, resulting in relations between the basic cohomology of the characteristic foliation and the number of closed Reeb orbits on an irregular K-cosymplectic manifold.
We investigate left-invariant Hitchin and hypo flows on $5$-, $6$- and $7$-dimensional Lie groups. They provide Riemannian cohomogeneity-one manifolds of one dimension higher with holonomy contained in $SU(3)$, $G_2$ and $Spin(7)$, respectively, which are in general geodesically incomplete. Generalizing results of Conti, we prove that for large classes of solvable Lie groups $G$ these manifolds cannot be completed: a complete Riemannian manifold with parallel $SU(3)$-, $G_2$- or $Spin(7)$-structure which is of cohomogeneity one with respect to $G$ is flat, and has no singular orbits. We furthermore classify, on the non-compact Lie group $SL(2,C)$, all half-flat $SU(3)$-structures which are bi-invariant with respect to the maximal compact subgroup $SU(2)$ and solve the Hitchin flow for these initial values. It turns out that often the flow collapses to a smooth manifold in one direction. In this way we recover an incomplete cohomogeneity-one Riemannian metric with holonomy equal to $G_2$ on the twisted product $SL(2,C)\times_{SU(2)} C^2$ described by Bryant and Salamon.
We prove that Riemannian foliations on complete contractible manifolds have a closed leaf, and that all leaves are closed if one closed leaf has a finitely generated fundamental group. Under additional topological or geometric assumptions we prove that the foliation is also simple.
We describe a generalization of GKM theory for actions of arbitrary compact connected Lie groups. To an action satisfying the non-abelian GKM conditions we attach a graph encoding the structure of the non-abelian 1-skeleton, i.e., the subspace of points with isotopy rank at most one less than the rank of the acting group. We show that the algebra structure of the equivariant cohomology can be read off from this graph. In comparison with ordinary abelian GKM theory, there are some special features due to the more complicated structure of the non-abelian 1-skeleton.
The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers, the dimensions of the basic Dolbeault cohomology groups, only depend on the isomorphism class of the underlying CR structure. Equivalently, we show that the basic Hodge numbers are invariant under deformations of type I. This result reduces their computation to the quasi-regular case. In the second part, we show a basic version of the Carrell-Lieberman theorem relating the basic Dolbeault cohomology of (M,F) to that of (C,F), where C is the union of closed leaves of F. As a special case, if F has only finitely many closed leaves, then the (p,q)-th basic Hodge numbers are zero for p not equal to q. Combining the two results, we show that if M admits a nowhere vanishing CR vector field with finitely many closed orbits, then the (p,q)-th basic Hodge numbers are zero for p not equal to q. As an application of these results, we compute then the basic Hodge numbers for deformations of homogeneous Sasakian manifolds.
We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, which is a generalization of equivariant formality for torus actions without fixed points. As a consequence, a generic component of the contact moment map is a perfect Morse-Bott function for the basic cohomology of the orbit foliation F of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of F is trivial in odd degrees, and its dimension equals the number of closed Reeb orbits. We characterize the K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM type theorem for K-contact manifolds, which allows us to calculate the equivariant cohomology algebra of K-contact manifolds in presence of the nonisolated GKM condition.
We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce as corollaries several results concerning the usual (de Rham) cohomology of $M$, such as the following obstruction to the existence of a cohomogeneity one action: if $M$ admits a cohomogeneity one action, then $χ(M)>0$ if and only if $H^{\rm odd}(M)=\{0\}$.
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, i.e., 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.
We show that for every symmetric space G/K of compact type with K connected, the K-action on G/K by left translations is equivariantly formal.
We show that the well-known fact that the equivariant cohomology 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 maximal isotropy rank. 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.
The basic cohomology of a Riemannian foliation on a complete manifold 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.
We study higher-order conservation laws of the non-linearizable elliptic Poisson equation $ \frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u) $ as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. The S^1--symmetry of the PDE leads to a normal form for the undifferentiated conservation law as well. We show that for higher-order conservation laws to exist, it is necessary that $f$ satisfies a linear second order ODE. In this case, an at most real two--dimensional space of new conservation laws in normal form appears at each even prolongation. When $f_{uu}=βf$ this upper bound is attained and the work of Pinkall and Sterling allows them to be written explicitly. We relate higher-order conservation laws to generalized symmetries of the exterior differential system by identifying their generating functions. This Noether correspondence provides the connection between conservation laws and the canonical Jacobi fields of Pinkall and Sterling.
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 neo-classical 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 G_2/SO(4) to which our theory applies.
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.
Motivated by a paper of Zirnbauer, we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the Lie theoretical viewpoint are introduced, e.g. geodesics, isometry groups and invariant metrics on Lie supergroups and homogeneous superspaces.