A pseudo-Riemannian manifold is said to be spacelike Jordan IP if the Jordan normal form of the skew-symmetric curvature operator depends upon the point of the manifold, but not upon the particular spacelike 2-plane in the tangent bundle at that point. We use methods of algebraic topology to classify connected spacelike Jordan IP pseudo-Riemannian manifolds of signature $(p,q)$, where $q\ge 11$, $p\le \frac{q-6}{4}$ and where the set $\{q, . . ., q+p\}$ does not contain a power of 2.
We present several principal bundles of embeddings of compact manifolds (with or without boundary) whose base manifolds are nonlinear Grassmannians. We study their infinite dimensional differential manifold structure in the Fréchet category. This study is motivated by the occurrence of such objects in the geometric Lagrangian formulation of free boundary continuum mechanics and in the study of the associated infinite dimensional dual pairs structures.
We use the Cartan representations of $SO(3)$ and $SU(3)$, and an irreducible 14-dimensional representation of $Sp(3)$ to construct certain totally geodesic submanifolds in "skew" position in the complex quadrics, the complex 2-Grassmannians and the quaternionic 2-Grassmannians.
We give a representation of canonical vector bundles over Grassmannian manifolds as non-compact affine symmetric spaces as well as their Cartan model in the group of the Euclidean motions.
A new generalization of Grassmannians, called ν-grassmannians, and a canonical super vector bundle over this new space, say Γ, are introduced. Then, constructing a Gauss supermap of a super vector bundle, the universal property of Γ is discussed. Finally, we generalize one of the main theorems of homotopy classification for vector bundles in supergeometry.
We prove that for any k greater or equal to 2, given a smooth compact k-dimensional manifold and a multiplicative k-1-gerbe on a Lie group, together with an integrable connection, there is a line bundle on the corresponding Beilinson-Drinfeld Grassmannian having the factorization property. We show that taking global sections of this line bundle we obtain a factorization algebra.
Y. J. Suh and H. Lee (Bull. Korean. Math. Soc. 47, 551-561 (2010)) characterized real hypersurfaces $M$ of type $B$ by the invariance of vector bundle $JTM^\perp$ under the shape operator and the orthogonality of $JTM^\perp$ and $\mathcal {J}TM^\perp$, where $TM^\perp$, $J$ and $\mathcal J$ are the normal bundle of $M$, Kähler structure and Quaternionic Kähler structure of $G_2({\mathbb{C}}^{m+2})$ respectively. In this paper, we characterize real hypersurfaces $M$ of type A by the invariance of the vector bundle $JTM^\perp$ under the shape operator with the Reeb vector field in $\mathcal {J}T…
Spectral bundles on Abelian varieties, complex projective spaces and Grassmannians
Xiamen University Malaysia（厦门大学马来西亚分校）；Institute of Mathematics, Academia Sinica（中央研究院数学研究所）；National Center for Theoretical Sciences（国家理论科学中心）；Department of Mathematics, National Taiwan University（台湾大学数学系）
A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of this bundle and a Laplace operator. We apply our main theorem, itself a generalization of a Theorem of Takahashi, to generalize the theory of do Carmo and Wallach and to describe the moduli space of harmonic maps satisfying the gauge and the Einstein-Hermitian conditions from a compact Rieannian manifold into a Grassmannian. As an application, several rigidity results are exihibited. In particular we generalize a rigidity theorem due to Calabi in the case of holomo…
The aim of these notes is to describe how to construct canonical bundles of moving frames and differential invariants for parametrized curves in Lagrangian Grassmannians, at least in the monotonic case. Such curves appear as Jacobi curves of sub-Riemannian extremals [1,2] . Originally this construction was done in [6,7], where it uses the specifics of Lagrangian Grassmannian. In later works [3,4] a much more general theory for construction of canonical bundles of moving frames for parametrized or unparametrized curves in the so-called generalized flag varieties was developed, so that the probl…
In their study of the representation theory of loop groups, Pressley and Segal introduced a determinant line bundle over an infinite dimensional Grassmann manifold. Mickelsson and Rajeev subsequently generalized the work of Pressley and Segal and in the process introduced for any p >=1 another infinite dimensional Grassmann manifold and a determinant line bundle defined over it. The construction of this determinant line bundle required the notion of a regularized determinant for bounded operators. In this note we specialize to the case p =2 and construct explicitly a connection on the correspo…
Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Ja…
The skew mean curvature flow (SMCF) is a natural generalization of the famous vortex filament equation. In this note, we show that the Gauss map of the SMCF satisfies a Schrödinger flow equation. In this regard, we explore the geometry of the oriented Grassmannian manifold explicitly by embedding it into the exterior product space.
A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geometry. It is obtained by translating a Grassmannian analogue of the Laplace operator to a certain tractor bundle with a specific tractor connection, which is not normal in general.
This paper gives an example of special Lagrangian manifold obtained from a hypersurface of a complex Grassmannian with vanishing first Chern class. The obtained manifold is a 1-torus bundle over the two dimensional real projective space. Such manifolds are interesting for mirror symmetry theory. Other examples of the same type are provided at the end of this article.
Let G(k,n) be the Grassmannian of oriented subspaces of dimension k of R^n with its canonical Riemannian metric. We study the energy of maps assigning to each P \in G(k,n) a unit vector normal to P. They are sections of a sphere bundle E_{k,n}^1 over G(k,n). The octonionic double and triple cross products induce in a natural way such sections for k=2, n=7 and k=3, n=8, respectively. We prove that they are harmonic maps into E_{k,n}^1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tan…
Narasihman and Ramanan proved that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to relate geometric properties of the classifying map to geometric properties of the pulled-back connection. More specifically, we describe conditions on the classifying map under which the pulled-back connection: (1) is fat (in the sphere bundle), (2) has a parallel curvature tenso…
Consider the principal $U(n)$ bundles over Grassmann manifolds $U(n)\rightarrow U(n+m)/U(m) \stackrelπ\rightarrow G_{n,m}$. Given $X \in U_{m,n}(\mathbb{C})$ and a 2-dimensional subspace $\mathfrak{m}' \subset \mathfrak{m} $ $ \subset \mathfrak{u}(m+n), $ assume either $\mathfrak{m}'$ is induced by $X,Y \in U_{m,n}(\mathbb{C})$ with $X^{*}Y = μI_n$ for some $μ\in \mathbb{R}$ or by $X,iX \in U_{m,n}(\mathbb{C})$. Then $\mathfrak{m}'$ gives rise to a complete totally geodesic surface $S$ in the base space. Furthermore, let $γ$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\…
For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^1$-bundle one obtains central extensions for certain groups of diffeomorphisms of $M$. We can realize $Gr_{m-2}(M)$ as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians $SGr_{2k}(M)$ as coadjoint orbits in the group…
Consider the principal $U(n)$ bundles over the dual of Grassmann manifolds $U(n)\ra U(n,m)/U(m) \stackrelπ\ra D_{n,m}$. Given a 2-dimensional subspace $\frakm' \subset \frakm $ $ \subset \mathfrak{u}(n,m), $ assume either $\frakm'$ is induced by $X,Y \in U_{m,n}(\bbc)$ with $X^{*}Y = μI_n$ for some $μ\in \bbr$ or by $X,iX \in U_{m,n}(\bbc)$. Then $\frakm'$ gives rise to a complete totally geodesic surface $S$ in the base space. Furthermore, let $γ$ be a piecewise smooth, simple closed curve on $S$ parametrized by $0\leq t\leq 1$, and $\wtγ$ its horizontal lift on the bundle $U(n) \ra π^{-1}(S)…
Asymptotic expansion of induced Grassmannian Chern forms and distribution of random degeneracy sets
Sabancı University（萨班奇大学）；Syracuse University（雪城大学）；Great Bay University（大湾区大学）；Universität zu Köln（科隆大学）；Institute of Mathematics ‘Simion Stoilow’, Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）
Let $M=Σ_1\times Σ_2$ be the product of two compact Riemannian manifolds of dimension $n\geq 2 $ and two, respectively. Let $Σ$ be the graph of a smooth map $f:Σ_1\mapsto Σ_2$, then $Σ$ is an $n$-dimensional submanifold of $M$. Let ${\frak G}$ be the Grassmannian bundle over $M$ whose fiber at each point is the set of all $n$-dimensional subspaces of the tangent space of $M$. The Gauss map $γ:Σ\mapsto \frak{G} $ assigns to each point $x\in Σ$ the tangent space of $Σ$ at $x$. This article considers the mean curvature flow of $Σ$ in $M$. When $Σ_1$ and $Σ_2$ are of the same non-negative curvatur…
In this article we lift Pestov's Identity on the tangent bundle of a Riemannian manifold $M$ to the bundle of $k$-tuples of tangent vectors. We also derive an integrated version and a restriction to the frame bundle $P^kM$ of $k$-frames. Finally, we discuss a dynamical application for the parallel transport on $\mathcal{G}_{or}^k(M)$, the Grassmannian of oriented $k$-planes of $M$.
