Search

We give a comprehensive account of Chern's Theorem that S^6 admits no omega-compatible almost complex structures. No claim to originality is being made, as the paper is mostly an expanded version of material already in the literature. This article extends the talks that both authors gave in Marburg during the conference "(Non)existence of complex structures on S^6" in April 2017., Misprints corrected

We would like to present the ideas of Makoto Matsumoto and Andras Rapcsak in the projective theory of Finsler spaces. Finally, we give some remarks using of Matsumoto–Rapcsak Theorem.

Three-dimensional Harvey-Lawson submanifolds were introduced in an earlier paper by Akbulut and Salur [1] , as examples of Lagrangian-type manifolds inside a G 2 manifold. In this paper, we discuss these as well as two other similar types of submanifolds of G 2 and S p i n ( 7 ) manifolds and their deformations. We first show that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G 2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. We then introduce a new class of Lagrangian-type four-dimensional submanifolds of M, call them RS-submanifolds, and prove that the space of deformations of a smooth, compact, orientable RS-submanifold of a G 2 manifold M can be identified with the space of closed 3-forms on RS. Finally, we describe an analogous setting for S p i n ( 7 ) manifolds by defining a new class of Lagrangian-type four-dimensional submanifolds of a S p i n ( 7 ) manifold N, which we call L-submanifolds. We show that the space of deformations of a smooth, compact, orientable L-submanifold of N can be identified with the space of closed 3-forms on L.

The generalization of Frobenius' theorem to foliations with singularities is usually attributed to Stefan and Sussmann, for their simultaneous discovery around 1973. However, their result is often referred to without caring much on the precise statement, as some sort of magic spell. This may be explained by the fact that the literature is not consensual on a unique formulation of the theorem, and because the history of the research leading to this result has been flawed by many claims that turned to be refuted some years later. This, together with the difficulty of doing proof-reading on this topic, brought much confusion about the precise statement of Stefan-Sussmann's theorem. This paper is dedicated to bring some light on this subject, by investigating the different statements and arguments that were put forward in geometric control theory between 1962 and 1994 regarding the problem of integrability of generalized distributions. We will present the genealogy of the main ideas and show that many mathematicians that were involved in this field made some mistakes that were successfully refuted. Moreover, we want to address the prominent influence of Hermann on this topic, as well as the fact that some statements of Stefan and Sussmann turned to be wrong. In this paper, we intend to provide the reader with a deeper understanding of the problem of integrability of generalized distributions, and to reduce the confusion surrounding these difficult questions., Comment: 16 pages, v4: final version

Let $ (E,h) $ be a Griffiths semipositive Hermitian holomorphic vector bundle of rank $ 3 $ over a complex manifold. In this paper, we prove the positivity of the characteristic differential form $ c_1(E,h) \wedge c_2(E,h) - c_3(E,h) $, thus providing a new evidence towards a conjecture by Griffiths about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, we establish a new chain of inequalities between Chern forms. Moreover, we point out how to obtain the positivity of the second Chern form $ c_2(E,h) $ in any rank, starting from the well-known positivity of such form if $ (E,h) $ is just Griffiths positive of rank $ 2 $. The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions., 16 pages, no figures, comments are very welcome! v4: some minor changes. A remark added in Section 2 following referee's comments. A reference has been added. Version accepted for publication in Differ. Geom. Appl

In this paper, we study left invariant Einstein metrics on compact semisimple Lie groups. A new method to construct holonomy irreducible non-naturally reductive Einstein metrics on certain compact semisimple (non-simple) Lie groups is presented. In particular, we show that if G is a classical compact simple Lie group and H is a closed subgroup such that G / H is a standard homogeneous Einstein manifold, then there exist holonomy irreducible non-naturally reductive Einstein metrics on H × G , except for some very special cases. A further interesting result of this paper is that for any compact simple Lie group G, there always exist holonomy irreducible non-naturally reductive Einstein metrics on the compact semisimple Lie groups G n , for any n ≥ 4 .

In this paper we review the well-known fact that the only spheres admitting an almost complex structure are S 2 and S 6 . The proof described here uses characteristic classes and the Bott periodicity theorem in topological K-theory. This paper originates from the talk “Almost Complex Structures on Spheres” given by the second author at the MAM1 workshop “(Non)-existence of complex structures on S 6 ”, held in Marburg from March 27th to March 30th, 2017. It is a review paper, and as such no result is intended to be original. We tried to produce a clear, motivated and as much as possible self-contained exposition.

The generalized conjugate connections on semi-Riemannian manifolds are studied in this paper. A fact that an affine connection is equiaffine iff it's conjugate connection is equiaffine on statistical manifolds was generalized to the case of generalized conjugate connections on semi-Riemannian manifolds in [3] . The facts that the conjugate symmetry and conjugate Ricci-symmetry of statistical manifolds are sufficient conditions for α-connections to be equiaffine for any α ( ∈ R ) are generalized to the case of generalized conjugate connections on semi-Riemannian manifolds in this paper.

Martingale optimal transport has attracted much attention due to its application in pricing and hedging in mathematical finance. The essential notion which makes martingale optimal transport different from optimal transport is peacock. A peacock is a sequence of measures which satisfies convex order property. In this paper we study peacock geodesics in Wasserstain space, and we hope this paper can provide some geometrical points of view to look at martingale optimal transport.

This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the bisections to a given Lie groupoid, where the second part is about the construction of Lie groupoids from candidates for their bisection Lie groups. The procedure of this second part becomes feasible due to some recent progress in the infinite-dimensional Frobenius theorem, which we heavily exploit. The main application to the prequantisation of (pre)symplectic manifolds comes from an integrability constraint of closed Lie subalgebras to closed Lie subgroups. We characterise this constraint in terms of a modified discreteness conditions on the periods of that manifold., 45 pages, LaTex, this is a sequel to arXiv:1409.1428

We found that we need to revise some proofs in the paper “Integral curvature bounds and bounded diameter with Bakry–Emery Ricci tensor”. Hence, we correct them in the present paper.

A fourth-order nonlinear dispersive partial differential equation arises in the field of mathematical physics, the solution of which is a curve flow on the two-dimensional unit sphere. In recent ten years, a geometric generalization of the sphere-valued physical model has been considered and the solvability of the initial value problem has been investigated. In particular, in the author's previous work, time-local existence and uniqueness result of the solution was established under the assumption that the solution is a closed curve flow on a compact Riemann surface with constant curvature. In the present paper, we propose a new geometric generalization of the sphere-valued physical model. As a main result, we show time-local existence of a solution to the initial value problem under the assumption that the solution is a closed curve flow on a compact locally Hermitian symmetric space. The proof is based on the geometric energy method combined with a gauge transformation to overcome the difficulty of the so-called loss of derivatives. Interestingly, the results can be applied to construct a generalized bi-Schrodinger flow proposed by Ding and Wang. The assumption on the manifold plays a crucial role both to enjoy a good solvable structure of the initial value problem and to reduce the generalized bi-Schrodinger flow equation to the one considered in the present paper.

We study Riemannian foliations whose transverse Levi-Civita connection $\nabla$ has special holonomy. In particular, we focus on the case where $Hol(\nabla)$ is contained either in SU(n) or in Sp(n). We prove a Weitzenbock formula involving complex basic forms on K\"ahler foliations and we apply this formula for pointing out some properties of transverse Calabi-Yau structures. This allows us to prove that links provide examples of compact simply-connected contact Calabi-Yau manifolds. Moreover, we show that a simply-connected compact manifold with a K\"ahler foliation admits a transverse hyper- K\"ahler structure if and only if it admits a compatible transverse hyper-Hermitian structure. This latter result is the "foliated version" of a theorem proved by Verbitsky. In the last part of the paper we adapt our results to the Sasakian case, showing in addition that a compact Sasakian manifold has trivial transverse holonomy if and only if it is a compact quotient of the Heisenberg Lie group., Comment: 19 pages. The paper includes, among some other original results, all the results contained in the e-print "On Sasakian manifolds with special transverse holonomy" arXiv:1204.2839

This paper is devoted to investigating the Mobius differential geometry of a new class of surfaces, named the surfaces with closed Mobius form. The main theorem shows that a surface with closed Mobius form can be determined by a smooth function satisfying a 5th order partial differential equation presented in this paper. As an application of the main theorem, the isothermic surfaces with closed Mobius form are classified.

The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through such a point in arbitrary tangential directions, but only in certain directions said to be admissible (the number of admissible directions is generically 1 or 3). Secondly, we study the global properties of geodesics in pseudo-Riemannian metrics possessing differentiable groups of symmetries. At the end of the paper, two special types of discontinuous metrics are considered., Comment: 21 pages, 14 figures

We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions., Comment: 17 papers. This paper contains parts of arXiv:1311.6749v1 which did not appear in the published version

In a previous paper we classified complete stationary surfaces (i.e. spacelike surfaces with zero mean curvature) in 4-dimensional Lorentz space R 1 4 which are algebraic and with total Gaussian curvature − ∫ K d M = 4 π . Here we go on with the study of such surfaces with − ∫ K d M = 6 π . It is shown in this paper that the topological type of such a surface must be a Mobius strip. On the other hand, new examples with a single good singular end are shown to exist.

The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $% \tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric $(\kappa,\mu)$-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric $(\kappa,\mu)$-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under $% \mathcal{D}$-homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed., Comment: Different. Geom. Appl. (to appear)

In this paper, we consider a special class of the surfaces in 3-sphere dened by oneparameter families of great circles. We give a generic classication of singularities of such surfaces and investigate the geometric meanings from the view point of spherical geometry. In this paper we investigate a special class of surfaces in 3-sphere which are called great circular surfaces. We say that a surface in 3-sphere is a great circular surface if it is given by a oneparameter family of great circles (cf., §4). On the other hand, there appeared two kinds of curvatures in the previous theory of surfaces in 3-sphere, One is called the extrinsic Gauss curvature Ke and another is the intrinsic Gauss curvature KI. The intrinsic Gauss curvature is nothing but the Gauss curvature defined by the induced Riemannian metric on the surface. The relation between these curvatures is known that Ke = KI − 1. We can show that an extrinsic flat surface is (at least locally) parametrized as a great circular surface (cf., Theorem 3.3). Such a surface is an extrinsic flat great circular surface (briefly, we call an E-flat great circular surface). This is one of the motivation to investigate great circular surfaces. In Euclidean space, surfaces with the vanishing Gauss curvature are developable surfaces which belong to a special class of ruled surfaces [5, 6]. Therefore, the notion of great circular surfaces is one of the analogous notions with ruled surfaces in 3-sphere. In this paper, we study geometric properties and singularities of great circular surfaces. However, there is the canonical double covering π : S 3 −→ RP 3 onto the projective space. A great circle corresponds to a projective line in RP 3 , so that the singularities of great circular surfaces are the same as those of ruled surfaces. There are a lot of researches on developable surfaces in R 3 ⊂ RP 3 from the view point of Projective differential geometry [2, 4, 12, 16]. We investigate the singularities of great circular surfaces from the view point of spherical geometry (i.e, SO(4)invariant geometry). For any smooth curve A : I −→ SO(4) in the rotation group SO(4), we can define a parametrization FA of a great circular surface M = Image FA in 3-sphere. We can easily show

We discuss the question whether a (complete) parallel submanifold M of a Riemannian symmetric space N is an (extrinsically) homogeneous submanifold, i.e. whether there exists a subgroup of the isometries of N which acts transitively on M . In a previous paper, we have discussed this question in case the universal covering space of M is irreducible. It is the subject of this paper to generalize this result to the case when the universal covering space of M has no Euclidian factor.

In a 1991 paper by Buttig and Eichhorn, the existence and uniqueness of a differential forms heat kernel on open manifolds of bounded geometry was proven. In that paper, it was shown that the heat kernel obeyed certain properties, one of which was a relationship between the derivative of heat kernel of different degrees. We will give a proof of this condition for complete manifolds with Ricci curvature bounded below, and then use it to give an integral representation of the heat kernel of degree $k$.

In the present paper we consider manifolds equipped with a paraconformal structure, understood as the tangent bundle isomorphic to a symmetric tensor product of rank-two vector bundles. If an ordinary differential equation satisfies Wunschmann condition then it defines a paraconformal structure on solution space. In the present paper we characterize all paraconformal structures which can be obtained in this way. In particular, we provide a new proof that all paraconformal structures on 3-dimensional manifolds are defined by ODEs. We show that if the dimension is greater than 3 then there exist structures which are not defined by an ODE.

We observed in our previous paper that all the complex structures on four-dimensional compact solvmanifolds, including tori, are left-invariant. In this paper we will give an example of a six-dimensional compact solvmanifold which admits a continuous family of non-left-invariant complex structures. We also provide a complete classification of three-dimensional compact homogeneous complex solvmanifolds; and determine which of them admit pseudo-Kahler structures.

The ‘traditional’ curve-straightening flow is based on one of the standard Sobolev inner products and it is known to break certain symmetries of reflection. The purpose of this paper is to show that there are alternative Riemannian structures on the space of curves that yield flows that preserve symmetries. This feature comes at a price. In one symmetrizing metric the gradient vector fields are considerably more demanding to compute. In another symmetrizing metric smoothness is lost. This investigation will also explain the phenomena of ‘spinning’ as observed in several examples in the traditional flow. Three classes of alternative Riemannian structures are examined. The first class includes the traditional metric as a special case and is shown to never preserve both rotation symmetries and symmetries of reflection. The second class consists of a single metric corresponding to one of the standard Sobolev metrics, and is shown to preserve both types of symmetries. The third class also includes the traditional metric but it is shown that there is a unique different metric in this class, which preserves both types of symmetries. This particular metric generally yields smooth vector fields, which when evaluated at a smooth function do not give a smooth element of the corresponding tangent space. The third class is nevertheless ‘preferred’ since it has the distinction that it ‘respects’ the projection induced by the derivative operator onto the tangent bundle of the space of derivatives. The paper concludes with a number of graphical illustrations that show preserved symmetry and removal of spinning.

In this paper we construct new examples of symmetric non-totally geodesic submanifolds in irreducible symmetric spaces of non-compact type and of rank⩾2. These symmetric spaces are characterized by the fact that they contain a reflective submanifold with one-dimensional Euclidean factor; they are listed at the end of the paper.

In the paper, we establish some conditions which ensure one of the following: (i) the existence of the pullback of the quantization bundle of a Poisson manifold to a quantization bundle of a symplectic leaf, (ii) the existence of the projection of a quantization bundle from a presymplectic realization of a Poisson manifold to the manifold or to its symplectic leaves. The main case is that of an isotropic realization. The paper ends by a discussion of the notion of a polarization of a Poisson manifold.

Let (M_1,F_1) and (M_2,F_2) be a pair of Finsler manifolds. The Minkowskian product Finsler manifold (M,F) of (M_1,F_1) and (M_2,F_2) with respect to a product function f is the product manifold M=M_1\times M_2 endowed with the Finsler metric F^2=f(K,H), where K=(F_1)^2,H=(F_2)^2. In this paper, the Cartan connection and Berwald connection of (M,F) are derived in terms of the corresponding objects of (M_1,F_1) and (M_2,F_2). Necessary and sufficient conditions for (M,F) to be Berwald (resp. weakly Berwald, Landsberg, weakly Landsberg) manifold are obtained. Thus an effective method for constructing special Finsler manifolds mentioned above is given.

In this paper, we characterize and classify the isoparametric hypersurfaces with constant principal curvatures in the product spaces $ \mathbb{Q}^{2}_{c_{1}} \times \mathbb{Q}^{2}_{c_{2}}$, where $\mathbb{Q}^{2}_{c_{i}}$ is a space form with constant sectional curvature $c_{i}$, for $c_1 \neq c_2$.

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 pseudoK\"ahler 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

Previous methods for exact linearization by feedback have relied on solving Frobenius systems of partial differential equations of dimensions equal to the Kronecker indices. We will describe an algorithm whereby one may find the linearizing feedback for any controlable linearizable system having distinct Kronecker indices with p -controls by purely algebraic calculations and integration of at most p one-dimensional Frobenius systems. The paper concludes with a concrete example considered by Hunt-Su-Meyer in their paper [3].

In this paper we report on some recent results about maximal surfaces in a Lorentzian product space of the form M 2 ◊ R1, where M 2 is a connected Riemannian surface and M 2 ◊R1 is endowed with the Lorentzian metric h,i = h,iM dt 2 . In particular, if the Gaussian curvature of M is non-negative, we establish new Calabi-Bernstein results for complete maximal surfaces immersed into M 2 ◊ R1 and for entire maximal graphs over a complete surface M. We also construct counterexamples which show that our Calabi-Bernstein results are no longer true without the hypothesis KM 0. Finally, we introduce two local approaches to our global results. We do not provide here with detailed proofs of our results. For further details, we refer the reader to the original papers Ref. 1‐3.

In this paper, we present a survey of some variational problems in geometry of submanifolds, which includes our recent research results in geometry of rminimal submanifolds, geometry of Willmore submanifolds and variations of some parametric elliptic functional. We also propose some open problems at the end of paper.

This paper reviews the grading technique developed by Getzler to prove the local index theorem and shows how to adapt it to compute the leading terms of asymptotic expansions of traces of heat kernels in two other situations., Comment: 16 pages

In this paper, we present new almost Schur type inequalities for weighted manifolds, under constraints on k-Bakry-Emery Ricci tensor. We also address results for weighted manifolds with convex boundary. The main tools for these results are a Pohozaev-Schoen type identity to weighted manifolds and the Reilly identity for the k-Bakry Emery tensor.

In this paper, inspired by Fernandez-Lopez and Garcia-Rio [11] , we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13] , we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.

In this paper, we establish a complete classification of locally strongly convex isotropic centroaffine hypersurfaces in the ( n + 1 ) -dimensional affine space R n + 1 .