We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A_8, is complete and non-singular on R^8. The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S^4, and are denoted by B_8^+, B_8^- and B_8. The metrics on B_8^+ and B_8^- occur in families with a non-trivial parameter. The metric on B_8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A_8. The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP^3. We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L^2-normalisable harmonic 4-form for the A_8 manifold, and two such 4-forms (of opposite dualities) for the B_8 manifold.
G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G2 solutions are Einstein of positive scalar curvature and are uniquely determined by the simple symmetry group. During the proof the equations for G2-symplectic and G2-cosymplectic structures are studied and the topological types of the manifolds admitting such structures are determined. New examples of c…
We classify all cohomogeneity one manifolds with principal orbit Q^111=SU(2)^3/U(1)^2 or M^110=(SU(3) x SU(2))/(SU(2) x U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically SU(4) and that they are asymptotically conical. Moreover, we investigate the smoothness of the metrics at the singular orbit.
We investigate cohomogeneity-one metrics whose principal orbit is an Aloff-Wallach space SU(3)/U(1). In particular, we are interested in metrics whose holonomy is contained in Spin(7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighborhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed b…
We clarify the global geometry of two 1-parameter families of cohomogeneity one Spin(7) holonomy metrics with generic orbit the Aloff--Wallach space $N(1,-1) \cong \mathrm{SU}(3)/\mathrm{U}(1)$ and singular orbits $S^5$ and $\mathbb{C}P^2$, which at short distance were shown to exist by Reidegeld. The two families fit into the geography of previously known families of cohomogeneity one metrics with exceptional holonomy and provide a Spin(7) analogue of the well-known conifold transition in the setting of Calabi--Yau 3-folds. Furthermore, we discover that there is another transition to families…
There is a rich theory of so-called (strict) nearly Kaehler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kaehler 6-manifold. A central problem in the field has been the absence of any complete …
We construct a continuous 1-parameter family of smooth complete Ricci-flat metrics of cohomogeneity one on vector bundles over $\mathbb{CP}^2$, $\mathbb{HP}^2$ and $\mathbb{OP}^2$ with respective principal orbits $G/K$ the Wallach spaces $SU(3)/T^2$, $Sp(3)/(Sp(1)Sp(1)Sp(1))$ and $F_4/\mathrm{spin}(8)$. Almost all the Ricci-flat metrics constructed have generic holonomy. The only exception is the complete $G_2$ metric discovered in [BS89][GPP90]. It lies in the interior of the 1-parameter family on $\bigwedge_-^2\mathbb{CP}^2$. All the Ricci-flat metrics constructed have asymptotically conical…
Most known four-dimensional cohomogeneity-one Einstein metrics are diagonal in the basis defined by the left-invariant one-forms, though some essentially non-diagonal ones are known. We consider the problem of explicitly seeking non-diagonal Einstein metrics, and we find solutions which in some cases exhaust the possibilities. In particular we construct new examples of neutral signature non--diagonal Bianchi type VIII Einstein metrics with self--dual Weyl tensor.
One way of producing explicit Riemannian 6-manifolds with holonomy SU(3) is by integrating a flow of SU(2)-structures on a 5-manifold, called the hypo evolution flow. In this paper we classify invariant hypo SU(2)-structures on nilpotent 5-dimensional Lie groups. We characterize the hypo evolution flow in terms of gauge transformations, and study the flow induced on the variety of frames on a Lie algebra taken up to automorphisms. We classify the orbits of this flow for all hypo nilpotent structures, obtaining several families of cohomogeneity one metrics with holonomy contained in SU(3). We p…
Let $G/H$ be a connected, simply connected homogeneous space of a compact Lie group $G$. We study $G$-invariant quasi-Einstein metrics on the cohomogeneity one manifold $G/H\times (0,1)$ imposing the so-called monotypic condition on $G/H$. We obtain estimates on the rate of blow-up for these metrics near a singularity under a mild assumption on $G/H$. Next, we demonstrate that we can find quasi-Einstein metrics satisfying arbitrary $G$-invariant Dirichlet conditions.
In this paper, we classify compact simply connected cohomogeneity one manifolds up to equivariant diffeomorphism whose isotropy representation by the connected component of the principal isotropy subgroup has three or less irreducible summands. The manifold is either a bundle over a homogeneous space or an irreducible symmetric space. As a corollary such manifolds admit an invariant metric with non-negative sectional curvature.
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 furthe…
Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when O is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKähler metrics with G-invariant Kähler potentials if and only if g is sp(3), su(6), so(7), so(12) or e_7 and otherwise is the unique G-invariant hyperKähler met…
We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G_2-ma…
A Kähler metric is called central if the determinant of its Ricci endomorphism is constant. For the case in which this constant is zero, we study on $4$-manifolds the existence of complete metrics of this type which are cohomogeneity one for three unimodular $3$-dimensional Lie groups: $SU(2)$, the group of Euclidean plane motions $E(2)$ and a quotient by a discrete subgroup of the Heisenberg group $\mathrm{nil}_3$. We obtain a complete classification for $SU(2)$, and some existence results for the other two groups, in terms of specific solutions of an associated ODE system.
This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Ziller. Two series of candidates arose in this classification for which it is not yet know if they admit positive curvature. The connection of these candidates to the Hitchin self dual Einstein orbifolds is discussed as well, including the curvature properties of the Hitchin metrics.
We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic metric at one of the ends, and at the other end the ball equipped with our metric collapses to a Poincaré-Einstein manifold of one lower dimension with an isolated conical singularity.
In these notes, after an introduction to toric Kahler geometry, we present Calabi's family of U(n)-invariant extremal Kahler metrics in symplectic action-angle coordinates and show that it actually contains, as particular cases, many interesting cohomogeneity one examples of constant scalar curvature.
A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential when the orbit is of cohomogeneity two. In some cases, we find that this structure lies in a one-parameter family of hyperKähler metrics with Kähler potentials, generalising the Eguchi-Hanson metrics in dimension four.
We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb C\mathbb P^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman, Gauduchon. Furthermore, we construct new examples …
Bryant and Salamon gave a construction of metrics of G2 holonomy on the total space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional self-dual Einstein manifold. We generalise it by considering the total space of an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection on this bundle. We make essentially the same ansatz for the calibrating 3-form, but use the curvature 2-forms instead of the ASD ones. We show that the resulting 3-form defines a metric of G2 holonomy if the connection satisfies a certain second-order PDE. This is exactly the same PDE that…
Let $G/H$ be a compact homogeneous space, and let $\hat{g}_0$ and $\hat{g}_1$ be $G$-invariant Riemannian metrics on $G/H$. We consider the problem of finding a $G$-invariant Einstein metric $g$ on the manifold $G/H\times [0,1]$ subject to the constraint that $g$ restricted to $G/H\times \{0\}$ and $G/H\times \{1\}$ coincides with $\hat{g}_0$ and $\hat{g}_1$, respectively. By assuming that the isotropy representation of $G/H$ consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.
We consider hypersurfaces in Einstein-Sasaki 5-manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5-dimensional metric from such a hypersurface, analogous to the (nearly) hypo and half-flat evolution equations in higher dimensions. We use these equations to classify Einstein-Sasaki 5-manifolds of cohomogeneity one.
In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds, that do not support invariant metrics of non-negative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case where both are equal to two, where existence of non-negatively curved metrics is known.
We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.
In this paper we discuss the smoothness conditions for metrics on a cohomogeneity one manifold, i.e. metrics invariant under a Lie group whose generic orbits are hypersurfaces. Along these hypersurfaces one describes the metrics in terms of a collection of functions defined along a geodesic normal to the hypersurfaces. In a neighborhood of a lower dimensional orbit the functions must satisfy certain smoothness conditions for the metric to extend smoothly. In general these can be quite complicated. We present a method that makes it straightforward to compute them, and illustrate it in several e…
We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds with positive curvature, apart from spheres, there are two infinite families among the 7-dimensional Eschenburg spaces and 13-dimensional Bazaikin spaces and one isolated 7-dimensional Berger space with this property. In addition to these examples, it turns out that only one isolated 7-manifold, and two infinite 7-dimensional families, potentially admit invaria…
We show that a certain family of cohomogeneity one manifolds does not admit an invariant metric of nonnegative sectional curvature, unless it admits one with positive curvature. As a consequence, the classification of nonnegatively curved cohomogeneity one manifolds in dimension 7 is reduced to only one further family of candidates
We show that integrability of an almost complex structure in complex dimension $m$ is equivalent, in the presence of an almost hermitian metric, to $m(m-1)$ equations involving what we call shear operators. Inspired by this, we give an ansatz for Kähler metrics in dimension $m>1$, for which at most $m-1$ of these shear equations are non-trivial. The equations for gradient Kähler-Ricci solitons in this ansatz are frame dependent PDEs, which specialize to ODEs under extra assumptions. Metrics solving the latter system include a restricted class of cohomogeneity one metrics, and we find among the…
We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in $\{2,3,4,6,8\}$. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.
In this article, we construct two continuous 1-parameter family of non-compact $\mathrm{Spin}(7)$ metrics with both chiralities, with the principal orbit an Aloff--Wallach space $N_{k,l}$ and the singular orbit $\mathbb{CP}^2$. For generic $N_{k,l}$, metrics constructed are locally asymptotically conical (ALC). For $N_{1,1}$, we construct two continuous 1-parameter families with geometric transition from asymptotically conical (AC) metrics to ALC metrics.
We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general formalism becomes a practical tool for calculating homogeneous or cohomogeneity one examples, and we illustrate this with some newly discovered cases of Spin(7) and G2 metrics.
We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $S^3 \times S^1$. We partition the class of such nearly-Ka}hler structures into three types (called I, II, III) and prove a local existence and generality result for each type. Metrics of Types I and II are shown to be incomplete. We also derive a quasilinear elliptic PDE system on the 2-dimen…
IAs is well known, when D6 branes wrap a special lagrangian cycle on a non compact CY 3-fold in such a way that the internal string frame metric is Kahler there exists a dual description, which is given in terms of a purely geometrical eleven dimensional background with an internal metric of $G_2$ holonomy. It is also known that when D6 branes wrap a coassociative cycle of a non compact $G_2$ manifold in presence of a self-dual two form strength the internal part of the string frame metric is conformal to the $G_2$ metric and there exists a dual description, which is expressed in terms of a pu…
A cohomogeneity one manifold is a manifold with the action of a compact Lie group, whose quotient is one dimensional. Such manifolds are of interest in Riemannian geometry, in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. In this paper we classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.
We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor space of any compact quaternion Kahler space is Kahler-Einstein. The reduction of the holonomy to a subgroup of Spin(7) gives non linear system relating three unknown functions of one variable. We obtain a particular solution and we find that the resulting metric is a Calabi-Yau …
In this paper we give a classification of cohomogeneity one manifolds admitting an invariant metric with quasipositive sectional curvature except for two $7$-dimensional families. The main result carries over almost verbatim from the classification results in positive curvature carried out by Verdiani and Grove, Wilking and Ziller. Three main tools used in the positively curved case that we generalized to quasipositively curved cohomogeneity one manifolds are Wilking's Chain Theorem, the classification of positively curved fixed point homogeneous manifolds by Grove and Searle and the Rank Lemm…
K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.
Consider a smooth manifold $M$. Let $G$ be a compact Lie group which acts on $M$ with cohomogeneity one. Let $Q$ be a singular orbit for this action. We study the gradient Ricci soliton equation $\Hess(u)+\Ric(g)+\fracε{2}g=0$ around $Q$. We show that there always exists a solution on a tubular neighbourhood of $Q$ for any prescribed $G$-invariant metric $g_Q$ and shape operator $L_Q$, provided that the following technical assumption is satisfied: if $P=G/K$ is the principal orbit for this action, the $K$-representations on the normal and tangent spaces to $Q$ have no common sub-representation…
We construct Yang-Mills connections on SO(n)-bundles over spheres equipped with the Euclidean metric. We use a cohomogeneity one group action on the bundle to reduce the Yang-Mills-equation to a system of ordinary differential equations. The system is shown to have solutions by variational methods, using ideas from harmonic map theory. Examples include Yang-Mills connections on each of the countably many principal SO(6)-bundles over $S^6$, and countably many Yang-Mills connections on $TS^n$ for $n\in\{5,...,9\}$.
We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory…
Let $M$ be a domain enclosed between two principal orbits on a cohomogeneity one manifold $M_1$. Suppose $T$ and $R$ are symmetric invariant (0,2)-tensor fields on $M$ and $\partial M$, respectively. The paper studies the prescribed Ricci curvature equation $\mathrm{Ric}(G)=T$ for a Riemannian metric $G$ on $M$ subject to the boundary condition $G_{\partial M}=R$ (the notation $G_{\partial M}$ here stands for the metric induced by $G$ on $\partial M$). Imposing a standard assumption on $M_1$, we describe a set of requirements on $T$ and $R$ that guarantee global and local solvability.
F. Podestà and A. Spiro introduced a class of $G$-manifolds $M$ with a cohomogeneity one action of a compact semisimple Lie group $G$ which admit an invariant Kaehler structure $(g,J)$ (``standard $G$-manifolds") and studied invariant Kaehler and Kaehler-Einstein metrics on $M$. In the first part of this paper, we gave a combinatoric description of the standard non compact $G$-manifolds as the total space $M_φ$ of the homogeneous vector bundle $M = G\times_H V \to S_0 =G/H$ over a flag manifold $S_0$ and we gave necessary and sufficient conditions for the existence of an invariant Kaehler-Eins…
In this article, we construct non-compact complete Einstein metrics on two infinite series of manifolds. The first series of manifolds are vector bundles with $\mathbb{S}^{4m+3}$ as principal orbit and $\mathbb{HP}^{m}$ as singular orbit. The second series of manifolds are $\mathbb{R}^{4m+4}$ with the same principal orbit. For each case, a continuous 1-parameter family of complete Ricci-flat metrics and a continuous 2-parameter family of complete negative Einstein metrics are constructed. In particular, $\mathrm{Spin}(7)$ metrics $\mathbb{A}_8$ and $\mathbb{B}_8$ discovered by Cvetič et al. in…
Let $M^n$, $n \in \{4,5,6\}$, be a compact, simply connected $n$-manifold which admits some Riemannian metric with non-negative curvature and an isometry group of maximal possible rank. Then any smooth, effective action on $M^n$ by a torus $T^{n-2}$ is equivariantly diffeomorphic to an isometric action on a normal biquotient. Furthermore, it follows that any effective, isometric circle action on a compact, simply connected, non-negatively curved four-dimensional manifold is equivariantly diffeomorphic to an effective, isometric action on a normal biquotient.
Non-compact G_2 holonomy metrics that arise from a T^2 bundle over a hyper-Kahler space are discussed. These are one parameter deformations of the metrics studied by Gibbons, Lu, Pope and Stelle in hep-th/0108191. Seven-dimensional spaces with G_2 holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational instantons are found, together with other examples. By considering the Apostolov-Salamon theorem math.DG/0303197, we construct a new example that, still being a T^2 bundle over hyper-Kahler, represents a non trivial two parameter deformation of the metrics studied in hep-th/010819…
We show that some embedded standard $13$-spheres in Shimada's exotic $15$-spheres have $\mathbb{Z}_2$ quotient spaces, $P^{13}$s, that are fake real $13$-dimensional projective spaces, i.e., they are homotopy equivalent, but not diffeomorphic to the standard $\mathbb{R}\mathrm{P}^{13}$. As observed by F. Wilhelm and the second named author in [RW], the Davis $\mathsf{SO}(2)\times \mathsf{G}_2$ actions on Shimada's exotic $15$-spheres descend to the cohomogeneity one actions on the $P^{13}$s. We prove that the $P^{13}$s are diffeomorphic to well-known $\mathbb{Z}_2$ quotients of certain Briesko…
