EXTREMAL RAYS AND NULL GEODESICS ON A COMPLEX CONFORMAL MANIFOLD

1994 ◽  
Vol 05 (01) ◽  
pp. 141-168 ◽  
Author(s):  
YUN-GANG YE
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Null Geodesic ◽  
Conformal Structure ◽  
Dimensional Manifold ◽  
Null Geodesics

A holomorphic conformal structure on a complex manifold X is an everywhere non-degenerate section [Formula: see text] for some line bundle N. In this paper, we show that if X is a projective complex n-dimensional manifold with non-numerically effective Kx and admits a holomorphic conformal structure, then X ≅ ℚn. This in particular answers affirmatively a question of Kobayashi and Ochiai. They asked if the same holds assuming c1 (X) > 0. As a consequence, we also show that any projective conformal manifold with an immersed rational null geodesic is necessarily a smooth hyperquadric ℚn.

A generalization of the Mariot theorem on congruences of null geodesics

1986 ◽  
Vol 405 (1828) ◽  
pp. 41-48 ◽  
Keyword(s):  
Vector Field ◽  
Conformal Structure ◽  
Dimensional Manifold ◽  
Hopf Fibration ◽  
Null Geodesics ◽  
Null Congruence ◽  
Dimension One

It is shown that the property of a congruence of curves to consist of null geodesics can be defined in terms of a distribution of a co-dimension one, without reference to the conformal structure of the underlying differen­tiable manifold: if k is the vector field tangent to the congruence and k is a 1-form characterizing the distribution, then the congruence is said to be null if k ˩ k = 0 and geodesic if, and only if, k ∧£ k = 0. The geodesic property of the congruence, on an n -dimensional manifold, means that if F is an ( n —2)-form such that k ˩ F = 0 and k ∧ F = 0, then k ∧ d F = 0. A twisting geodesic null congruence on S 1 ᵡ S 2 l + 1 , associated with the Hopf fibration S 2 l + 1 → CP l , is constructed as an illustration.

scholarly journals On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

2017 ◽  
Vol 19 (04) ◽  
pp. 1750043 ◽  
Author(s):  
Silvia Sabatini
Keyword(s):  
Fixed Points ◽  
Complex Manifold ◽  
Index Formula ◽  
Circle Action ◽  
Hilbert Polynomial ◽  
Dimensional Manifold ◽  
Hamiltonian Action ◽  
Almost Complex Manifold ◽  
Almost Complex ◽  
Chern Numbers

Let [Formula: see text] be a compact, connected, almost complex manifold of dimension [Formula: see text] endowed with a [Formula: see text]-preserving circle action with isolated fixed points. In this paper, we analyze the “geography problem” for such manifolds, deriving equations relating the Chern numbers to the index [Formula: see text] of [Formula: see text]. We study the symmetries and zeros of the Hilbert polynomial, which imply many rigidity results for the Chern numbers when [Formula: see text]. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a 6-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. Another consequence is that, if the action is Hamiltonian, the minimal Chern number coincides with the index and is at most [Formula: see text].

scholarly journals CONFORMAL COURANT ALGEBROIDS AND ORIENTIFOLD T-DUALITY

2012 ◽  
Vol 10 (02) ◽  
pp. 1250084 ◽  
Author(s):  
DAVID BARAGLIA
Keyword(s):  
Line Bundle ◽  
Bilinear Form ◽  
Conformal Structure ◽  
Crossed Module ◽  
Courant Algebroids ◽  
Courant Algebroid ◽  
Circle Bundles ◽  
Twisted Cohomology ◽  
Free Involution ◽  
Special Case

We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant algebroids and show they are classified by pairs (L, H) with L a flat line bundle and H ∈ H3(M, L) a degree 3 class with coefficients in L. As a special case gerbes for the crossed module (U(1) → ℤ2) can be used to twist TM ⊕ T*M into a conformal Courant algebroid. In the exact case there is a twisted cohomology which is 4-periodic if L2 = 1. The structure of Conformal Courant algebroids on circle bundles leads us to construct a T-duality for orientifolds with free involution. This incarnation of T-duality yields an isomorphism of 4-periodic twisted cohomology. We conjecture that the isomorphism extends to an isomorphism in twisted KR-theory and give some calculations to support this claim.

scholarly journals Characterisation theorems for compact hypercomplex manifolds

1987 ◽  
Vol 43 (2) ◽  
pp. 231-245
Author(s):  
S. Nag ◽  
J. A. Hillman ◽  
B. Datta
Keyword(s):  
Projective Space ◽  
Finite Number ◽  
Riemann Surfaces ◽  
Conformal Structure ◽  
Dimensional Manifold ◽  
Real Projective Space ◽  
Geometric Methods ◽  
Connected Surface ◽  
Compact Riemann Surfaces ◽  
Simply Connected

AbstractWe have defined and studied some pseudogroups of local diffeomorphisms which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these ‘Further pseudogroups’ turns out to be a quaternionic (respectively octonionic) manifold.We characterise compact Further manifolds as being products of compact Riemann surfaces with appropriate dimensional spheres. It then transpires that a connected compact quaternionic (H) (respectively O) manifold X, minus a finite number of circles (its ‘real set’), is the orientation double covering of the product Y × P2, (respectively Y×P6), where Y is a connected surface equipped with a canonical conformal structure and Pn is n-dimensonal real projective space.A corollary is that the only simply-connected compact manifolds which can allow H (respectively O) structure are S4 and S2 × S2 (respectively S8 and S2×S6).Previous authors, for example Marchiafava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.

scholarly journals Stable Higgs bundles over positive principal elliptic fibrations

2018 ◽  
Vol 5 (1) ◽  
pp. 195-201
Author(s):  
Indranil Biswas ◽  
Mahan Mj ◽  
Misha Verbitsky
Keyword(s):  
Vector Bundle ◽  
Line Bundle ◽  
Complex Manifold ◽  
Higgs Bundle ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Higgs Bundles ◽  
Stable Vector Bundle ◽  
Hermitian Metric ◽  
The Third

AbstractLet M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

EXAMPLES OF A NON-VANISHING CONJECTURE OF KAWAMATA

2007 ◽  
Vol 18 (05) ◽  
pp. 527-533
Author(s):  
YU-LIN CHANG
Keyword(s):  
Line Bundle ◽  
Complex Manifold ◽  
Symmetric Spaces ◽  
Sufficient Conditions ◽  
Compact Complex Manifold ◽  
Holomorphic Line Bundle ◽  
Compact Type ◽  
Hermitian Symmetric Spaces ◽  
Canonical Line Bundle ◽  
Positive Holomorphic Line Bundle

Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H0(M, K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow–Siu [10] surfaces, Kähler threefolds given by Deraux [3] and examples of Zheng [17].

scholarly journals Covariance group for null geodesic expansion calculations, and its application to the apparent horizon

2021 ◽  
Author(s):  
Stephen L. Adler
Keyword(s):  
Coordinate System ◽  
Apparent Horizon ◽  
Null Geodesic ◽  
Product Formula ◽  
Null Geodesics ◽  
Covariance Group

We show that the recipe for computing the expansions [Formula: see text] and [Formula: see text] of outgoing and ingoing null geodesics normal to a surface admits a covariance group with nonconstant scalar [Formula: see text], corresponding to the mapping [Formula: see text], [Formula: see text]. Under this mapping, the product [Formula: see text] is invariant, and thus the marginal surface computed from the vanishing of [Formula: see text], which is used to define the apparent horizon, is invariant. This covariance group naturally appears in comparing the expansions computed with different choices of coordinate system.

SUB-WEYL GEOMETRY AND ITS LINEAR CONNECTIONS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250047 ◽  
Author(s):  
OANA CONSTANTINESCU ◽  
MIRCEA CRASMAREANU
Keyword(s):  
Tangent Bundle ◽  
Natural Extension ◽  
Conformal Structure ◽  
Point Of View ◽  
Dimensional Manifold ◽  
Finsler Spaces ◽  
Weyl Geometry ◽  
Cartan Form ◽  
Linear Connections ◽  
Local Expression

The aim of this paper is to study from the point of view of linear connections the data [Formula: see text] with M a smooth (n+p)-dimensional real manifold, [Formula: see text] an n-dimensional manifold semi-Riemannian distribution on M, [Formula: see text] the conformal structure generated by g and W a Weyl substructure: a map [Formula: see text] such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as one-form the Cartan form.

scholarly journals Submanifolds ofF-structure manifold satisfyingFK+(−)K+1F=0

2001 ◽  
Vol 26 (3) ◽  
pp. 167-172
Author(s):  
Lovejoy S. Das
Keyword(s):  
Positive Integer ◽  
Complex Manifold ◽  
Dimensional Manifold ◽  
Invariant Submanifold ◽  
Almost Complex Manifold ◽  
Complementary Distribution ◽  
Almost Complex ◽  
Integrability Conditions ◽  
Invariant Submanifolds ◽  
Fixed Positive Integer

The purpose of this paper is to study invariant submanifolds of ann-dimensional manifoldMendowed with anF-structure satisfyingFK+(−)K+1F=0andFW+(−)W+1F≠0for1<W<K, whereKis a fixed positive integer greater than2. The case whenKis odd(≥3)has been considered in this paper. We show that an invariant submanifoldM˜, embedded in anF-structure manifoldMin such a way that the complementary distributionDmis never tangential to the invariant submanifoldψ(M˜), is an almost complex manifold with the inducedF˜-structure. Some theorems regarding the integrability conditions of inducedF˜-structure are proved.

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