Classifies simply-connected pluriclosed manifolds with parallel Bismut torsion.
problem Classifying specific types of manifolds with parallel Bismut torsion.
method Complete classification through mathematical analysis.
result Established a splitting theorem for certain manifolds.
Characterizes Hermitian manifolds with Bismut parallel torsion.
problem Understanding curvature properties of Hermitian manifolds with specific torsion.
method Analyzes Bismut curvature tensor and parallel torsion conditions.
result Characterizes Bismut torsion parallel manifolds using Bismut curvature alone.
The study characterizes Hermitian manifolds with parallel Bismut-Strominger torsion.
problem Characterizing Hermitian manifolds with specific torsion properties.
method Analyzing the curvature tensor and properties of the Bismut-Strominger connection.
result A necessary and sufficient condition for Bismut torsion parallel manifolds.
Study on balanced Hermitian threefolds with parallel Bismut torsion.
problem Characterizing compact, balanced BTP threefolds.
method Detailed description of all compact, balanced BTP threefolds.
result Characterization of all compact, balanced BTP threefolds.
Study SKT and CYT manifolds with parallel Bismut torsion.
problem Characterize and construct compact complex manifolds with specific geometric properties.
method Characterization via universal cover, construction using mapping torus, investigation of generalized Kaehler structures.
result Existence of non-Bismut flat examples and characterization of universal covers.
Researchers classify invariant Hermitian structures on flag manifolds with parallel Bismut torsion.
problem Classifying invariant Hermitian structures with specific torsion properties on flag manifolds.
method Detailed analysis of invariant Hermitian structures on flag manifolds, proving conditions for parallel Bismut torsion.
result Conditions for the existence of parallel Bismut torsion on most flag manifolds.
Study on BAS manifolds with parallel torsion and curvature.
problem Characterizing and classifying BAS manifolds.
method Canonical reduction theorem, classification in homogeneous settings, construction of combined geometries.
result Classification of BAS manifolds in various settings.
The study characterizes compact homogeneous manifolds with Bismut parallel torsion.
problem Characterizing compact homogeneous manifolds with specific geometric properties.
method Investigating Hermitian manifolds with Bismut parallel torsion, focusing on locally homogeneous manifolds.
result Characterization of compact Chern flat BTP manifolds and properties of BTP compact Hermitian locally homogeneous manifolds.
The paper confirms a conjecture for Bismut torsion parallel metrics.
problem The existence of metrics with constant holomorphic sectional curvature on non-Kähler manifolds.
method Investigation of Bismut torsion parallel metrics.
result The conjecture is confirmed for all non-balanced BTP manifolds.
New insights on ACYT 6-manifolds reveal instanton conditions for curvature.
problem Understanding instanton conditions on ACYT 6-manifolds.
method Analyzing curvature and torsion properties of ACYT 6-manifolds.
result Curvature is an SU(3)-instanton if and only if torsion is parallel. Study on Hermitian metrics on Lie algebras with specific ideals.
problem Classifying Hermitian metrics on Lie algebras with abelian ideals.
method Examined unimodular Lie algebras with abelian ideals of codimension two, classified metrics.
result Classification of Bismut Kähler-like and Bismut torsion-parallel metrics.
We find Einstein metrics on homogeneous HKT manifolds.
problem Characterizing and finding Einstein metrics on homogeneous HKT manifolds.
method Characterization and construction of invariant HKT-Einstein metrics.
result Existence and uniqueness of invariant HKT-Einstein metrics.
In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. I…
New construction method for special geometric structures.
problem Creating strong HKT and generalized hyperkähler structures.
method Mapping tori construction of skew-symmetric torsion.
result Non-product structures with parallel skew-symmetric torsion.
The paper confirms a conjecture about Hermitian manifolds with constant mixed curvature.
problem Compact Hermitian manifolds with non-zero constant mixed curvature must be Kähler.
method Verification for specific types of Hermitian manifolds including complex nilmanifolds, solvmanifolds, and Lie algebras.
result Partial evidence supporting Kai Tang's conjecture.
The paper computes a residue density for a specific Laplacian on compact manifolds.
problem Computing a noncommutative residue density for a complex Laplacian.
method Explicit computation of the noncommutative residue density associated with equivariant twisted Bismut Laplacian with torsion.
result Proves equivariant twisted Kastler-Kalau-Walze type theorems with torsion on compact manifolds with boundary.
Verify conjecture for special Hermitian manifolds.
problem Conjecture about space forms for canonical metric connections.
method Verify conjecture for complex nilmanifolds and Bismut torsion-parallel manifolds.
result Verify conjecture for two special types of Hermitian manifolds.
The Streets-Tian conjecture is confirmed for specific types of Hermitian manifolds.
problem The Streets-Tian conjecture on compact Hermitian manifolds.
method Elementary approach, explicit descriptions, and pathways of deformation.
result The conjecture is confirmed for special types of compact Hermitian manifolds.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
Study torsion and curvature in ACYT and AHKT 8-manifolds.
problem Characterize torsion and curvature in ACYT and AHKT 8-manifolds.
method Analyzes properties of Nijenhuis tensors, Ricci tensors, and torsion forms on ACYT and AHKT 8-manifolds.
result Closed torsion condition and Ricci flatness for ACYT 8-manifolds.
Study Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.
problem Characterize Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.
method Analyze SKT condition and Bismut Kähler-like metrics, construct new examples, and study pluriclosed flow.
result Construct new examples of Hermitian manifolds satisfying Bismut Kähler-like condition.
For a complex flat vector bundle over a fibered manifold, we consider the 1-parameter family of certain deformed sub-signature operators introduced by Ma-Zhang. We compute the adiabatic limit of the Bismut-Freed connection associated to this family and show that the Bismut-Lott analytic torsion form shows up naturally …
The paper compares two torsion invariants in complex vector bundles.
problem Comparing two torsion invariants in complex vector bundles.
method Constructing Bismut-Lott analytic torsion classes and showing they coincide with Igusa-Klein torsions.
result Bismut-Lott analytic torsion classes coincide with Igusa-Klein torsions for trivial flat line bundles.
We investigate the deformation theory of a class of generalized calibrations in Riemannian manifolds for which the tangent bundle has reduced structure group U(n), SU(n), G_2 and Spin(7). For this we use the property of the associated calibration form to be parallel with respect to a metric connection which may have no…
In this paper, we prove a conjecture raised by Angella, Otal, Ugarte, and Villacampa recently, which states that if the Strominger connection (also known as Bismut connection) of a compact Hermitian manifold is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler man…
The paper studies harmonic complex structures and special metrics on Sasakian manifolds.
problem Analyzing harmonic complex structures and special metrics on products of Sasakian manifolds.
method Examining 2-parameter families of Hermitian structures (Ja,b,ga,b) and their properties. result The complex structure Ja,b is harmonic with respect to ga,b, and conditions for various types of special metrics are determined. New insights prevent certain types of metrics on compact spaces.
problem Preventing the existence of specific types of metrics on compact spaces.
method Computing cohomology and analyzing stability of metrics for the pluriclosed flow.
result Prevents the existence of non-flat homogeneous Bismut Hermitian Einstein metrics on C-spaces.
Paper compares higher torsions and removes fiberwise Morse function assumption.
problem Comparing higher torsions from analytic and topological perspectives.
method Introduced fiberwise generalized Morse functions (GMFs) and excised neighborhoods around birth-death points.
result Established a generalized version of the higher Cheeger-Müller/Bismut-Zhang theorem.
Analytic surgery and gluing formula for torsion forms in fiber bundles.
problem Behavior of torsion forms under analytic surgery in fiber bundles.
method Analytic surgery and gluing formula for Bismut-Lott torsion and eta forms.
result Gluing formula for Bismut-Lott analytic torsion and eta forms under surgery limit.
Study on compact strong HKT manifolds and their properties.
problem Characterizing the structure of compact strong HKT manifolds.
method Geometric analysis, rigidity theorems, classification, and properties of Ricci foliations.
result Compact strong HKT manifolds are Hopf fibrations over compact 4-dimensional orbifolds.
Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component containing a unitary representation the Bismut-Zhang formula follows rather easi…
The paper examines SKT and CYT metrics on Lie groups.
problem Existence and properties of SKT and CYT metrics on Lie groups.
method Study left-invariant metrics on compact semi-simple Lie groups.
result Left-invariant metrics that are both CYT and SKT must be Bismut flat.
We compare the higher analytic torsion of Bismut and Lott of a fibre bundle p: M -> B equipped with a flat vector bundle F -> M and a fibre-wise Morse function h on M with a higher torsion T that is constructed in terms of a families Thom-Smale complex associated to h and F, thereby extending previous joint work with B…
New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
Study on instantons in G2 and Spin(7) manifolds.
problem Investigate instanton properties in G2 and Spin(7) manifolds. method Analyze the characteristic and torsion connections on G2 and Spin(7) manifolds, focusing on parallel torsion and Lee forms. result Conditions for the curvature of the characteristic connection to be a G2 instanton and for the torsion connection to be a Spin(7) instanton are established. We generalize a theorem of Bismut-Zhang, which extends the Cheeger-Mueller theorem on Ray-Singer torsion and Reidemeister torsion, to the case where the flat vector bundle over a closed manifold carries a nondegenerate symmetric bilinear form. As a consequence, we prove the Burghelea-Haller conjecture which gives an an…
A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on natural…
We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments.
Classifies Riemannian manifolds with specific torsion properties.
problem Classifying Riemannian manifolds with parallel, non-twistorial torsion.
method Classifies complete simply connected Riemannian manifolds with a metric connection having parallel torsion, non-zero vectorial component, and zero twistorial component.
result Classifies complete simply connected Riemannian manifolds with the specified torsion properties.
About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recen…
The study shows how certain geometries can be mapped to simpler structures.
problem Understanding the structure of geometries with specific properties.
method Reframing known submersion results and applying them to new contexts.
result Holonomy decompositions and holomorphic submersions are derived for specific geometries.
Let X be a compact connected strongly pseudoconvex CR manifold of dimension 2n+1,n≥1 with a transversal CR S1-action on X. We introduce the Fourier components of the Ray-Singer analytic torsion on X with respect to the S1-action. We establish an asymptotic formula for the Fourier components of the an…
Unified view of geometries with parallel skew torsion via submersions.
problem No de Rham decomposition for geometries with torsion.
method Developed and unified submersion constructions for geometries with parallel skew torsion.
result Completed and extended classification of irreducible geometries with parallel skew torsion.
Equivalence proven between two torsion invariants for flat vector bundles.
problem Equivalence of Igusa-Klein and Bismut-Lott torsion invariants for flat vector bundles.
method Reduction to trivial flat line bundles using Artin's induction theorem.
result Igusa-Klein and Bismut-Lott torsion invariants are equivalent for flat vector bundles with finite holonomy.
We present a new proof, as well as a C/Q extension, of the Riemann-Roch-Grothendieck theorem of Bismut-Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of η-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott a…
Classifies 7D manifolds with specific geometric properties.
problem Classifying 7D manifolds with parallel skew-symmetric torsion and G2 holonomy. method Extending Friedrich's work, using classification techniques for naturally reductive spaces and nearly parallel G2-structures. result Complete classification of 7D manifolds with the specified properties.
Study proves Kählerness criteria for Hermitian surfaces under specific curvature conditions.
problem Determining when Hermitian surfaces are Kählener.
method Used explicit identities linking Strominger-Bismut Ricci curvatures to torsion, and Chern number identities.
result Proves several Kählerness criteria for compact Hermitian surfaces.
Connection, torsion and curvature are introduced for general (local) Leibniz algebroids. Generalized Bismut connection on TM⊕ΛpT∗M is an example leading to a scalar curvature of the form R+H2 for a closed (p+2)-form H.