We characterize compact locally conformal parallel (respectively, ) manifolds as fiber bundles over with compact nearly Kähler (respectively, compact nearly parallel ) fiber. A more specific characterization is provided when the local parallel structures are flat.
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48 results for “compact 7-manifolds”
The paper studies curvature identities and solitons on Spin(7)-manifolds.
problem Curvature identities and solitons on Spin(7)-manifolds.
method Analyzes the curvature and torsion of Spin(7)-manifolds, proving identities and conditions.
result Conditions for closed torsion and implications for Ricci flatness and solitons.
New -instantons constructed on Joyce's manifold.
problem Constructing -instantons on Joyce's compact manifold.
method Gluing non-flat connections on local model spaces to a flat connection on the -orbifold.
result More than 20,000 new four-parameter families of -instantons.
Real analyticity proven for G_2 structures on compact 7-manifolds.
problem Analyzing real analyticity of G_2 structures on compact manifolds.
method Laplacian flow approach to study G_2 structures.
result Real analyticity of for each fixed positive time .
Researchers create a compact 7-manifold with unique structure and first Betti number 1.
problem Constructing a compact manifold with specific structure properties.
method Constructing a formal 7-manifold with a closed -structure and first Betti number 1, and demonstrating non-existence of torsion-free structures.
result The construction of a compact manifold with a closed -structure and first Betti number 1, which does not admit any torsion-free -structure.
New submanifolds identified in Spin(7) manifolds with identified deformation spaces.
problem Identifying deformations of Lagrangian-type submanifolds in Spin(7) manifolds.
method Defining L-submanifolds and showing their deformation spaces are closed 3-forms.
result Space of deformations of L-submanifolds identified as closed 3-forms.
Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.
problem Understanding smooth fibrations of compact Spin(7)-manifolds by Cayley submanifolds.
method Geometric and topological constraints from Spin(7)-structure, spinnability criterion, gauge-theoretic input.
result Rules out smooth Cayley fibrations on all known compact torsion-free Spin(7)-manifolds.
Constructs Spin(7) manifolds from self-dual Einstein 4-orbifolds.
problem Creating complete non-compact Spin(7) manifolds.
method Analytic construction using adiabatic limits of Spin(7) metrics on Seifert bundles over G2 orbifolds.
result Produces complete non-compact Spin(7) manifolds with arbitrary second Betti numbers and infinitely many families of ALC metrics.
New Spin(7) manifolds created from Calabi-Yau bundles.
problem Creating complete non-compact Spin(7) manifolds.
method Building on -bundles over AC Calabi-Yau manifolds.
result First known examples of complete toric Spin(7) manifolds.
Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…
New Spin(7)-manifolds constructed via generalized connected sums.
problem Creating compact Spin(7)-manifolds for geometric engineering.
method Generalized connected sum of Calabi-Yau four-fold and G2-holonomy manifold.
result Construction of new compact Spin(7)-manifolds with holonomy.
We find a necessary and sufficient condition for a compact 7-manifold to admit a -structure. As a result we find a sufficient condition for an open 7-manifold to admit a closed 3-form of -type.
Found a new compact G2-structure on a 7-manifold.
problem Identifying compact G2-structures on manifolds.
method Lattice in solvable Lie group, Laplacian coflow analysis.
result Obtained a 4-parameter subfamily of expanding solitons.
We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU…
Deformations of singular Cayley submanifolds studied.
problem Constructing fibrations of compact Spin(7) manifolds.
method Deformation theory of conically singular and asymptotically conical Cayley submanifolds.
result Detailed description of the deformation theory.
Survey of G2-holonomy manifold constructions.
problem Finding compact Riemannian 7-manifolds with G2 holonomy.
method Two methods: resolving orbifolds and gluing asymptotically cylindrical pieces.
result Examples of compact G2-holonomy manifolds constructed.
We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces …
Desingularizes conically singular Cayley submanifolds.
problem Constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds.
method Desingularization through gluing rescaled asymptotically conical submanifolds.
result Conically singular Cayley submanifolds can be desingularized.
Study on G2-manifolds shows automorphism group is abelian.
problem Understanding automorphism groups of G2-manifolds.
method Analyzing compact 7-manifolds with closed non-parallel G2-structures.
result The identity component of automorphism group is abelian with dimension bounded.
Study on instantons in and manifolds.
problem Investigate instanton properties in and manifolds.
method Analyze the characteristic and torsion connections on and manifolds, focusing on parallel torsion and Lee forms.
result Conditions for the curvature of the characteristic connection to be a instanton and for the torsion connection to be a instanton are established.
We construct examples of exponentially asymptotically cylindrical Riemannian 7-manifolds with holonomy group equal to G_2. To our knowledge, these are the first such examples. We also obtain exponentially asymptotically cylindrical coassociative calibrated submanifolds. Finally, we apply our results to show that one of…
We calculate the integer cohomology ring and stable tangent bundle of a family of compact, 3-Sasakian 7-manifolds constructed by Boyer, Galicki, Mann, and Rees. Previously only the rational cohomology ring was known. The most important part of the cohomology ring is a torsion group that we describe explicitly and whose…
The abstract describes a class of eternal solutions to the G-Laplacian flow.
problem Investigating properties of G-structures on compact 7-manifolds.
method Explicit description and investigation of the G-Laplacian flow starting from an extremally Ricci-pinched closed G-structure.
result The solution exists for all real times and remains extremally Ricci-pinched.
The paper finds CSC Sasaki metrics on specific 7-manifolds.
problem Finding constant scalar curvature Sasaki metrics.
method Using Boothby-Wang construction and weighted extremal approach.
result Explicit CSC Sasaki metrics on certain 7-manifolds.
New examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds found.
problem Identifying non-formal Sasaki-Einstein 7-manifolds and their submanifolds.
method Construction of new examples and analysis of fibre bundles, total spaces, and Sasaki-Einstein structures.
result Examples of non-formal Sasaki-Einstein 7-manifolds and their submanifolds.
We give a differential-geometric construction of compact manifolds with holonomy which is based on Joyce's second construction of compact -manifolds in \cite{Joyce00} and Kovalev's gluing construction of -manifolds in \cite{Kovalev03}. We also give some examples of compact $\ma…
We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Rieman…
In this note we construct a first example of a closed 3-form of -type on . We prove that does not admit a homogeneous 3-form of -type. Thus our example is a first example of a closed 3-form of -type on a compact 7-manifold which is not stably homogeneou…
We study generalized Killing spinors on compact Einstein manifolds with positive scalar curvature. This problem is related to the existence compact Einstein hypersurfaces in manifolds with parallel spinors, or equivalently, in Riemannian products of flat spaces, Calabi-Yau, hyperkaehler, G_2 and Spin(7) manifolds.
New flow on 7-torus yields torsion-free G2 structure.
problem Finding all-time existence and convergence of G2 flow.
method Hypersymplectic flow on standard torus.
result First all-time converging G2 flow on compact 7-manifold.
We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of t…
Constructs harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.
problem Analyzing harmonic 1-forms on K3-fibred Calabi-Yau 3-folds.
method Analytic construction of nowhere-vanishing harmonic 1-forms.
result Produces examples of compact 7-manifolds with holonomy G2.
Study classifies certain 7-manifolds with specific cohomology and proves positive Ricci curvature.
problem Classifying 1-connected 7-manifolds with specific cohomology rings.
method Analyzing cohomology rings and proving properties of metrics.
result Closed, smooth, spin, 1-connected 7-manifolds with specific cohomology admit metrics with positive Ricci curvature.
The study classifies 1-connected 7-manifolds with torsion-free second homology.
problem Classifying 1-connected 7-manifolds with specific homological properties.
method Generalizes a previous result using Kreck-Stolz invariants and quadratic refinements.
result New classification method for 2-connected and simply connected 7-manifolds.
The abstract discusses constructing Sasaki-Einstein metrics on specific 7-manifolds.
problem Finding Sasaki-Einstein metrics on certain 7-manifolds.
method Explicit construction of Sasaki-Einstein metrics on specified 7-manifolds.
result Explicit construction of Sasaki-Einstein metrics on a class of 7-manifolds.
Study calculates deformations of instantons on a specific -manifold.
problem Analyzing instantons on a specific -manifold.
method Deformation theory developed by the author.
result Virtual dimensions of moduli spaces of instantons calculated.
New -holonomy manifolds from 5d N=1 theories domain walls.
problem Geometrizing domain walls in 5d N=1 theories.
method Constructing 7-manifolds by fibering a Calabi-Yau over a real line.
result 7-manifolds with -holonomy from domain walls in 5d theories.
Study on harmonic forms on special holonomy manifolds, proving vanishing results.
problem Analyzing harmonic forms on complete special holonomy manifolds.
method Examined harmonic forms on and manifolds with nonzero parallel forms.
result Vanishing of harmonic 2-forms on and manifolds.
Constructs 7-manifolds with SO(3) invariant non-negative curvature.
problem Finding exotic spheres with non-negative curvature.
method Six-parameter family of highly connected 7-manifolds constructed with invariant metrics.
result All exotic spheres in 7 dimensions have non-negative curvature.
Study cohomotopy sets of simply connected 7-manifolds using suspension decompositions.
problem Understanding cohomotopy sets of simply connected 7-manifolds.
method Establish homotopy decompositions of the reduced suspension space into simpler spaces localized at primes.
result Established homotopy decompositions leading to insights into cohomotopy sets.
We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This rec…
Study -manifolds with a 4-torus action using a symmetric matrix ansatz.
problem Characterize -manifolds with a 4-torus action.
method Provide a Gibbons-Hawking type ansatz using a symmetric -matrix of functions.
result First known -manifolds with a rank 4 symmetry group and full holonomy.
Study connects -structures to flat connections on compact 3-manifolds.
problem Understanding moduli spaces of -structures and flat connections.
method Equivalence between moduli spaces of -structures and flat connections on compact 3-manifolds.
result Equivalence between moduli spaces of -structures and flat connections on compact 3-manifolds.
Method constructs G_2-instantons using twisted connected sums.
problem Constructing G_2-instantons over compact manifolds.
method Applying gluing result to instantons over 7-manifolds with tubular ends.
result Explicit understanding of deformation theory and asymptotic behaviour of constructed instantons.
The paper identifies conditions for free circle actions on specific 7-manifolds.
problem Determining conditions for free circle actions on certain 7-manifolds.
method Analyzing specific 7-manifolds and their components.
result Identifies conditions for free circle actions on .
Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.
problem Analyzing -structures on Calabi-Yau 7-manifolds.
method Reduced Ansätze for the Laplacian coflow on various Calabi-Yau 7-manifolds.
result Obtained a modified Kähler-Ricci flow.
Study instantons on asymptotically conical Spin(7)-manifolds, identifying deformation spaces.
problem Deformation theory of instantons on specific Spin(7)-manifolds.
method Relating deformation complex to spinors, identifying kernel of twisted negative Dirac operator.
result Virtual dimension of moduli space calculated using index theorem and Dirac operator spectrum.
New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.
problem Proving some hyperbolic manifolds don't virtually fiber.
method Hyperbolic reflection group trick, embedding theory, manifold topology.
result Constructed Gromov hyperbolic 7-manifolds that don't virtually fiber over a circle.