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13 pages, 272 KB

Open AccessArticle

Some Remarks on Existence of a Complex Structure on the Compact Six Sphere

by Daniel Guan, Na Li and Zhonghua Wang

Axioms 2024, 13(10), 719; https://doi.org/10.3390/axioms13100719 - 17 Oct 2024

Viewed by 1576

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on

S^{6}

was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a [...] Read more.

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on

S^{6}

was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on

S^{6}

. We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem. Full article

(This article belongs to the Section Geometry and Topology)

34 pages, 558 KB

Open AccessArticle

D-Branes in Para-Hermitian Geometries

by Vincenzo Emilio Marotta and Richard J. Szabo

Universe 2022, 8(4), 200; https://doi.org/10.3390/universe8040200 - 23 Mar 2022

Cited by 6 | Viewed by 1901

Abstract

We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by maximally isotropic vector bundles which do not generally admit the standard geometric picture in terms of submanifolds. When reduced to the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold, integrable branes yield D-branes as leaves of foliations which are interpreted as Dirac structures on the physical spacetime. We define a notion of generalised para-complex D-brane, which realises our D-branes as para-complex versions of topological A/B-branes. We illustrate how our formalism recovers standard D-branes in the explicit example of reductions from doubled nilmanifolds. Full article

(This article belongs to the Special Issue Dualities and Geometry)

12 pages, 327 KB

Open AccessArticle

On the Real Homotopy Type of Generalized Complex Nilmanifolds

by Adela Latorre, Luis Ugarte and Raquel Villacampa

Mathematics 2020, 8(9), 1562; https://doi.org/10.3390/math8091562 - 11 Sep 2020

Cited by 1 | Viewed by 2044

Abstract

We prove that for any

n \geq 4

, there are infinitely many real homotopy types of

2 n

-dimensional nilmanifolds admitting generalized complex structures of every type k, for

0 \leq k \leq n

. Full article

(This article belongs to the Section B: Geometry and Topology)

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