We use some natural lifts defined on the cotangent bundle T*M of a Riemannian manifold (M,g) in order to construct an almost Hermitian structure (G,J) of diagonal type. The obtained almost complex structure J on T*M is integrable if and only if the base manifold has constant sectional curvature and the coefficients as well as their derivatives, involved in its definition, do fulfill a certain algebraic relation. Next one obtains the condition that must be fulfilled in the case where the obtained almost Hermitian structure is almost Kaehlerian. Combining the obtained results we get a family of Kaehlerian structures on T*M, depending on two essential parameters. Next we study three conditions under which the considered Kaehlerian structures are Einstein. In one of the obtained cases we get that (T*M,G,J) has constant holomorphic curvature.
We obtain a locally symmetric Kaehler Einstein structure on a tube in the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained Kaehler Einstein structure cannot have constant holomorphic sectional curvature.
We obtain a class of Kaehler Einstein structures on the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained class of Kaehler Einstein structure depends on one essential parameter, cannot have constant holomorphic sectional curvature and is not locally symmetric.
We obtain a class of locally symetric Kaehler Einstein structures on the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained class of Kaehler Einstein structures depends on one essential parameter and cannot have constant holomorphic sectional curvature.
We obtain a Kaehler Einstein structure on the nonzero cotangent bundle of a Riemannian manifold of positive constant sectional curvature. The obtained Kaehler Einstein structure cannot have constant holomorphic sectional curvature and is not locally symmetric.
We obtain a locally symmetric Kaehler Einstein structure on the cotangent bundle of a Riemannian manifold of negative constant sectional curvature. Similar results are obtained on a tube around zero section in the cotangent bundle, in the case of a Riemannian manifold of positive constant sectional curvature. The obtained Kaehler Einstein structures cannot have constant holomorphic sectional curvature.
We obtain a class of locally symmetric Kaehler Einstein structures on the cotangent bundle of a Riemannian manifold of negative sectional curvature. Similar results are obtained in the case of a Riemannian manifold of positive sectional curvature. The obtained class of Kaehler Einstein structures depends on one essential parameter and cannot have constant holomorphic sectional curvature.
We use the natural lifts of the fundamental tensor field g to the cotangent bundle T*M of a Riemannian manifold (M,g), in order to construct an almost Hermitian structure (G,J) of diagonal type on T*M. The obtained almost complex structure J on T*M is integrable if and only if the base manifold has constant sectional curvature and the second coefficient, involved in its definition is expressed as a rational function of the first coefficient and its first order derivative. Next one shows that the obtained almost Hermitian structure is almost Kaehlerian. Combining the obtained results we get a f…
We construct a Kaehler structure on the punctured cotangent bundle of the Cayley projective plane whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and we show that the geodesic flow action is holomorphic and is expressed in a quite explicit form. We also give an embedding of the punctured cotangent bundle of the Cayley projective plane into the space of 8 times 8 complex matrices.
We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.
It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.
We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem for the natural diagonal (almost) para-Kählerian structures on the total spaces of the cotangent bundle.
We generalize reduction theorems for classical connections to operators with values in $k$-th order natural bundles. Using the first reduction theorem in order two we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
Starting with an orientable compact real-analytic Riemannian manifold $(L,g)$ with $χ(L)=0$, we show that a small neighbourhood $ \textrm{Op}(L) $ of the zero section in the cotangent bundle $T^{*}L$ carries a Calabi-Yau structure such that the zero section is an isometrically embedded special Lagrangian submanifold.
Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy equivalent, the corresponding Poisson algebras are homotopy equivalent. We apply this result to $L_\infty$-algebroids to show that two homotopy equivalent bundles have the same $L_\infty$-algebroid structures and explore some consequence on the theory of shifted Poisson structures.
We study prolongation of Norden structures on manifolds to their generalized tangent bundles and to their cotangent bundles. In particular, by using methods of generalized geometry, we prove that the cotangent bundle of a complex Norden manifold $(M,J,g)$ admits a structure of Norden manifold, $(T^{\star}(M),\tilde J, \tilde g)$. Moreover if $(M,J,g)$ has flat natural canonical connection then $\tilde J$ is integrable, that is $(T^{\star}(M),\tilde J, \tilde g)$ is a complex Norden manifold. Finally we prove that if $(M,J,g)$ is Kähler Norden flat then $(T^{\star}(M),\tilde J, \tilde g)$ is Kä…
A hyperkaehler manifold with a circle action fixing just one complex structure admits a natural a hyperholomorphic line bundle. This forms the basis for the construction of a corresponding quaternionic Kaehler manifold in the work of A.Haydys. We construct in this paper the corresponding holomorphic line bundle on twistor space and compute many examples, including monopole and Higgs bundle moduli spaces. We also show that the bundle on twistor space has a natural meromorphic connection which realizes it as the quantum line bundle for the hyperkaehler family of holomorphic symplectic structures…
Semisimple (co)adjoint orbits through real hyperbolic elements are well-known to be symplectomorphic to cotangent bundles. We provide a new proof of this fact based on elementary results on both Lie theory and symplectic geometry. Our proof establishes a new connection between the Iwasawa horospherical projection and the symplectic geometry of real hyperbolic (co)adjoint orbits.
In this paper the classification of left-invariant Riemannian metrics, up to the action of the automorphism group, on cotangent bundle of (2n+1)-dimensional Heisenberg group is presented. Also, it is proved that the complex structure on that group is unique and the corresponding pseudoKähler metrics are described and shown to be Ricci flat. It is well known that this algebra admits an ad-invariant metric of neutral signature. Here, the uniqueness of such metric is proved
This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kaehler-Ricci solitonic metric if and only if it is Fano.
An immersion $ϕ\colon M \to \tilde M$ of a manifold $M$ into an indefinite Kaehler manifold $\tilde M$ is called purely real if the almost complex structure $J$ on $\tilde M$ carries the tangent bundle of $M$ into a transversal bundle. In this article we survey some recent results on purely real surfaces in Kaehler surfaces as well as on Lorentz surfaces in Lorentzian Kaehler surfaces.
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie algebras and compact Lie algebras we recover by simple methods interesting co-homological known results. The Lie algebra of prederivations encompasses the one of derivations as a subalgebra. We find out that Lie algebras of cotangent Lie groups (which are not semi-simple) of sem…
We study the conditions under which a Kählerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved in the condition for $(T^*M,G,J)$ to be a Kählerian manifold, is expressed as a rational function of the other two, their derivatives, the constant sectional curvature of the base manifold $(M,g)$, and the constant holomorphic sectional curvature of the general natural Kählerian structure $(G,J)$.
Let $(M,g)$ be an n-dimensional Riemannian manifold and $T^{*}M$ be its cotangent bundle equipped with a Riemannian metric of Cheeger Gromoll type which rescale the horizontal part by a nonzero differentiable function. The main purpose of the present paper is to discuss curvature properties of $T^{*}M$ and construct almost paracomplex Norden structures on $T^{*}M$. We investigate conditions for these structures to be para-Kähler (paraholomorphic) and quasi-Kähler. Also, some properties of almost paracomplex Norden structures in context of almost product Riemannian manifolds are presented.
In this paper we study the geometrical structures on the cotangent bundle using the notions of adapted tangent structure and regular vector fields. We prove that the dynamical covariant derivative on $T^{*}M$ fix a nonlinear connection for a given $\mathcal{J}$-regular vector field. Using the Legendre transformation induced by a regular Hamiltonian, we show that a semi-Hamiltonian vector field on $T^{*}M$ corresponds to a semispray on $TM$ if and only if the nonlinear connection on $TM$ is just the canonical nonlinear connection induced by the regular Lagrangian.
In [SW2], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifol…
We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a Kählerian structure $(G,J)$ of general natural lift type (see \cite{Druta1}), is Einstein. We first obtain a general natural Kähler-Einstein structure on the cotangent bundle $T^*M$. In this case, a certain parameter, $λ$ involved in the condition for $(T^*M,G,J)$ to be a Kählerian manifold, is expressed as a rational function of the other two, the value of the constant sectional curvature, $c$, of the base manifold $(M,g)$ and the constant $ρ$ involved in the condition for the str…
We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of Cavalcanti-Gualtieri and Fernández-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a…
We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poisson structure from M to T*M via connections gives such bivector fields and we discuss the conditions for these lifts to be Poisson bivector fields and their compatibility with the canonical Poisson structure on T*M. Finally, for a 2-form on a Riemannian manifold, we study the cond…
We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds. As an application we have that a natural circle bundle over the Kuranishi moduli space of a Calabi-Yau threefold is a Lorentzian proper affine hypersphere.
We prove a Chekanov-type theorem for the spherization of the cotangent bundle $ST^*B$ of a closed manifold $B$. It claims that for Legendrian submanifolds in $ST^*B$ the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies.
This encyclopedia article briefly reviews without proofs some of the main results in cotangent bundle reduction. The article recalls most the necessary prerequisites to understand the main results.
We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that any discrete sub group of a Lie group of rank >1 that is not of Hodge type can't be the fundamental group of a quasi-compact Kaehler manifold.
In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmid's Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kähler manifolds with carefully controlled asymptotics near the compactifying divisor; such a metric is unique up to some isometry. Such an asymptotic behavior is canonical in some sense.
Let G be compact Lie group. It is shown that the cotangent bundle of the complexification of G admits a hyperkahler structure which is invariant under left and right translations by elements of G. The proof is to realize the cotangent bundle of the complex group as a moduli space of solutions to Nahm's equations on the closed interval.
In a previous article, we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about the reduction of cotangent bundles.
We construct some lift of an almost complex structure to the cotangent bundle, using a connection on the base manifold. This generalizes the complete lift defined by I.Sato and the horizontal lift introduced by K.Yano and S.Ishihara. We study some geometric properties of this lift and its compatibility with symplectic forms on the cotangent bundle.
In this paper, the natural foliations in cotangent bundle T*M of Cartan space (M,K) are studied. It is shown that the geometry of these foliations is closely related to the geometry of the Cartan space (M,K) itself. This approach is used to obtain new characterizations of Cartan spaces with negative constant curvature.
We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.
A canonical hyperkaehler metric on the total space $T^*M$ of a cotangent bundle to a complex manifold $M$ has been constructed recently by the author (see alg-geom/9710026). This paper presents the results of alg-geom/9710026 in a streamlined and simplified form. The only new result is an explicit formula obtained for the case when $M$ is an Hermitian symmetric space.
We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is quasi-projective we prove that F is actually a Kaehler form.
We give topological obstructions to the existence of a closed exact Lagrangian submanifold in the cotangent bundle of a closed manifold M which is the total space of a fibration over the circle. For instance we show that the fundamental group of such a Lagrangian submanifold cannot be the free product of two non-trivial groups and that in any finite presentation of this group the difference between the number of generators and the number of relations is less than two.
A universal lower bound for the first positive eigenvalue of the Dirac operator on a compact quaternionic Kaehler manifold M of positive scalar curvature is calculated. It is shown that it is equal to the first positive eigenvalue on the quaternionic projective space. For this, the horizontal tangent bundle on the canonical SO(3)-bundle over M is equipped with a hyperkaehlerian structure and the corresponding splitting of the horizontal spinor bundle is considered. The desired estimate is obtained by looking at hyperkaehlerian twistor operators on horizontal spinors.
On a cotangent bundle $T\sp*G$ of a Lie group $G$ one can describe the standard Liouville form $θ$ and the symplectic form $d θ$ in terms of the right Maurer Cartan form and the left moment mapping (of the right action of $G$ on itself), and also in terms of the left Maurer-Cartan form and the right moment mapping, and also the Poisson structure can be written in related quantities. This leads to a wide class of exact symplectic stuctures on $T\sp*G$ and to Poisson structures by replacing the canonical momenta of the right or left actions of $G$ on itself by arbitrary ones, followed by reducti…
This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to all proper cotangent-lifted actions, around points with fully-isotropic momentum values. We also present a ``tangent-level'' commuting reduction result and use it to characterise the symplectic normal space of any cotangent-lifted action. In two special cases, we arrive at splitti…
Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.
For a connected Lie group G, we show that a complex structure on the total space TG of the tangent bundle of G that is left invariant and has the property that each left translation G-orbit is a totally real submanifold is induced from a smooth immersion of TG into the complexification of G. For G compact and connected, we then characterize left invariant and biinvariant complex structures on the total space T*G of the cotangent bundle of G which combine with the tautological symplectic structure to a Kaehler structure.
We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stein homotopic to the standard Stein structure on $T^*\mathbb{R}^4 = \mathbb{R}^8$. We use this to show that any exotic $\mathbb{R}^4$ embeds in the standard symplectic $\mathbb{R}^8$ as a Lagrangian submanifold. As a corollary, we show that $\mathbb{R}^8$ has uncountably many smoot…
Let K be a compact semi-simple Lie group. We classify K-invariant Kaehler structures on the space Kc/(P,P), where Kc is the complexification of K, P is a parabolic subgroup of Kc, and (P,P) the commutator subgroup. For each Kaehler structure, we study its moment map and associated pre-quantum line bundle for geometric quantization. Some holomorphic sections of the line bundle form a unitary K-representation space, and we study the multiplicity of its irreducible subrepresentations.
We prove a theorem on singular symplectic cotangent bundle reduction in the Fréchet setting and apply it to Yang-Mills-Higgs theory with special emphasis on the Higgs sector of the Glashow-Weinberg-Salam model. For the latter model we give a detailed description of the reduced phase space and show that the singular structure is encoded in a finite-dimensional Lie group action.
We discuss Lagrangian and Hamiltonian field theories that are invariant under a symmetry group. We apply the polysymplectic reduction theorem for both types of field equations and we investigate aspects of the corresponding reconstruction process. We identify the polysymplectic structures that lie at the basis of cotangent bundle reduction and Routh reduction in this setting and we relate them by means of the Routhian function and its associated Legendre transformation. We end the paper with examples that illustrate the applicability of our results.
