Bismut Einstein metrics on complex manifolds are Kähler Einstein or Bismut Ricci flat.
problem Characterizing Bismut Einstein metrics on compact complex manifolds.
method Observing the (2,0)-part of Bismut Ricci form and using it to prove properties of the metrics.
result Bismut Einstein metrics with non-zero Einstein constant are Kähler Einstein, and those with zero are Bismut Ricci flat.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
In this note we observe that on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure the (1,1)-part of the Bismut-Ricci form is seminegative definite. As application we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long…
New metrics found on homogeneous spaces, preserving key geometric properties.
problem Finding Bismut Ricci flat metrics on compact homogeneous spaces.
method Exhibited and proved uniqueness of Bismut Ricci flat metrics on homogeneous spaces.
result Unique and non-homothetic metrics found on certain homogeneous spaces.
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Study of complex 3-folds with vanishing Bismut Ricci form via special Kähler geometry.
problem Characterizing non-Kähler BHE 3-folds with vanishing Bismut Ricci form.
method Dimensional reduction to special Kähler geometry, solving 6th order PDE, momentum map interpretation.
result Characterization of Samelson locally homogeneous BHE 3-folds and construction of new non-Kähler BHE structures.
The paper proves stability of a Ricci flat metric on a product of Einstein homogeneous spaces.
problem Stability of Bismut Ricci flat metrics on product spaces.
method Generalized Ricci flow on aligned homogeneous spaces.
result The Bismut Ricci flat metric is asymptotically and globally stable under the generalized Ricci flow.
We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for stu…
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. The paper classifies Bismut Kähler-like manifolds in dimensions 4 and 5.
problem Classifying Bismut Kähler-like manifolds in specific dimensions.
method Structural theorems and proving conjectures about BKL manifolds.
result Complete classifications of BKL manifolds in dimensions 4 and 5.
Study bi-Hermitian metrics on complex surfaces and solve geometric PDEs.
problem Construct canonical metrics on complex surfaces with split tangent bundle.
method Introduced new fully non-linear geometric PDEs and established smooth solutions.
result Solved the prescribed Bismut Ricci problem on complex surfaces.
Solves generalized Kähler Calabi-Yau problem on compact manifolds.
problem Calabi conjecture in generalized Kähler geometry.
method New local deformation result, Bismut Ricci curvature transgression formula, generalized Kähler-Ricci flow.
result Global existence and convergence of flow for initial data in generalized Kähler class of Kähler Calabi-Yau structure.
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.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
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.
New metrics found on non-Kähler complex manifolds.
problem Finding metrics on non-Kähler complex manifolds.
method Reinterpretation of Bismut Hermitian-Einstein condition and associated holomorphic Courant algebroid.
result Infinitely many non-Kähler manifolds without Bismut Hermitian-Einstein metrics.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Constructing Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Using t-Gauduchon metrics on principal torus bundles over rational homogeneous varieties. result Examples of new metrics on non-Kähler Calabi-Yau 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.
New interpretation of complex hyperbolic form as Weil-Petersson form.
problem Understanding complex hyperbolic structures on moduli spaces.
method Interpreting complex hyperbolic form as a Weil-Petersson form for punctured spheres.
result Found a new equality between two symplectic forms.
Evolutionary forms, as well as exterior forms, are skew-symmetric differential forms. But in contrast to the exterior forms, the basis of evolutionary forms is deforming manifolds (with unclosed metric forms). Such forms possess a peculiarity, namely, the closed inexact exterior forms are obtained from that. The closur…
If a closed 3-manifold M supports a closed, nonsingular, irrational 1-form which linearly deforms into contact forms, then M supports a K-contact form. On the 3-torus, a closed nonsingular 1-form deforms linearly into contact forms if and only if it is a fibration 1-form. on any other 2-torus bundle over the circle, ev…
The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the properties and specific features of the differential-geometrical structures. The…
The study examines parallel forms on manifolds, focusing on specific dimensions and forms.
problem Characterizing parallel forms with constant components in various dimensions.
method Analyzing forms in dimensions 6 and n, providing geometric characterizations.
result The converse implication holds for (n-2)-forms and 3-forms in dimension 6, but fails for certain exceptional cases.
The study shows that the second fundamental form is intrinsic under certain conditions in space forms.
problem Understanding the intrinsic nature of the second fundamental form in space forms.
method Proving the intrinsic nature of the normalized second fundamental form A under specific conditions. result The normalized second fundamental form A is intrinsic if σ2k+1(A)eq0 for some k≥1. New forms generalize Whitney forms with rational coefficients for numerical analysis.
problem Numerical problems with singularities near simplex faces.
method Introduce shadow forms and degrees of freedom for integration over faces of blow-up simplices.
result Obtain isomorphism between shadow forms cohomology and cellular cohomology of blow-up simplices.
This paper classifies quadratic form parameters over integers and computes their Witt groups.
problem Classifying quadratic form parameters over integers and computing their Witt groups.
method Study of quadratic forms and extended quadratic forms over the integers, defining and comparing different definitions of extended quadratic forms.
result Classification of all quadratic form parameters over the integers and computation of their Witt groups.
Abstract: Generalizes multisymplectic forms to vector-valued versions.
problem Generalizing multisymplectic forms to vector-valued versions.
method Obtained a standard local presentation and proved an entropy inequality for partial compositions.
result Vector-valued multisymplectic forms form a non-unital operad.
The paper finds a contact form on SL(2p) for p > 1.
problem Left invariant Pfaffian forms on SL(2p) are not contact forms for p > 1.
method Constructed a contact form invariant under SO(2p).
result Found a contact form on SL(2p) for p > 1.
Researchers solve conformal Killing forms on Kaehler manifolds.
problem Classifying conformal Killing forms on compact Kaehler manifolds.
method Explicit determination of conformal Killing forms in middle degree.
result First examples of conformal Killing forms not from Hamiltonian 2-forms.
Characterizes Whitney forms on simplices and proves their uniqueness.
problem Characterizing Whitney forms on simplices.
method Proves the uniqueness of differential forms with affine coefficients.
result Whitney forms are the unique differential forms with affine coefficients.
Study of tautological forms on curve moduli spaces.
problem Understanding tautological forms on moduli spaces of curves.
method Defined and studied a system of tautological rings on moduli spaces of marked curves, showing certain 2-forms are tautological and rings are finite dimensional.
result Characterized the Kawazumi-Zhang invariant as a tautological form.
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.
Paper presents a new triangular form for flat systems.
problem Designing flat systems with two inputs.
method Geometric characterization and static feedback equivalence.
result Sufficient condition for affine input systems to be flat.
Special p-forms are forms which have components φ_{μ_1...μ_p} equal to +1,-1 or 0 in some orthonormal basis. A p-form φ\in Λ^p R^d is called democratic if the set of nonzero components {φ_{μ_1...μ_p}} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1,...,d}. Knowledge of these symmetry …
Paper transforms torse-forming vector fields into simpler forms.
problem Generalizing vector fields and their transformations.
method Present techniques to transform torse-forming vector fields into simpler cases.
result Concrete examples of transformations are provided.
The study finds Lagrangian submanifolds in adjoint semisimple orbits for real forms.
problem Characterizing Lagrangian submanifolds in adjoint semisimple orbits.
method Analyzing real flags and orbits of real forms with respect to symplectic forms.
result Classification of infinitesimally tight Lagrangian submanifolds in the compact case and Lagrangian submanifolds in the complex case.
Study on immersions with flat normal bundle in curved spaces.
problem Behavior of isometric immersions with negative curvature.
method Investigation of second fundamental form growth in space forms.
result Second fundamental form grows exponentially if normal bundle is flat.
Paper studies second order symmetric parallel tensors in generalized f.pk-space forms.
problem Exploring properties of second order symmetric parallel tensors in generalized f.pk-space forms.
method Analyzes the properties of second order symmetric parallel tensors and deduces the existence or non-existence of certain tensors and hypersurfaces.
result There does not exist second order skew-symmetric parallel tensor in f.pk-space form. There is no parallel hypersurface in a generalized f.pk-space form but there is semi-parallel hypersurface.
We study conformal Killing forms on compact 6-dimensional nearly Kähler manifolds. Our main result concerns forms of degree 3. Here we give a classification showing that all conformal Killing 3-forms are linear combinations of dω and its Hodge dual ∗dω where ω is the fundamental 2-form of the nearly Kähler stru…
Diffeology explores k-forms and bundles with more information than traditional differential forms.
problem Understanding k-forms and bundles in diffeological spaces. method Developed theory of diffeological vector pseudo-bundles, including limits and colimits, and various operations.
result Sections of bundles of k-forms contain more information than differential forms. New conformally invariant forms help identify Einstein metrics.
problem Identifying Einstein metrics in conformal classes.
method Constructing new conformally invariant one-forms.
result Global obstructions to the existence of Einstein metrics.
In this paper, we first define the equivariant infinitesimal η-form, then we compare it with the equivariant η-form, modulo exact forms, by a locally computable form. As a consequence, we obtain the singular behavior of the equivariant η-form, modulo exact forms, as a function on the acting Lie group. This result…
Differential forms on the Fréchet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a const…
New metrics produce discrete zero sets for nondegenerate harmonic forms.
problem Creating metrics to produce discrete zero sets for nondegenerate harmonic forms.
method Metric perturbation to produce new nondegenerate harmonic forms with discrete zero sets.
result Existence of metrics producing discrete zero sets for nondegenerate harmonic forms.
Enhances power of covariance matrix tests for high-dimensional data.
problem Testing large covariance matrices in high-dimensional data.
method Proposes a new Fisher's combined probability test for quadratic form and maximum form statistics.
result Boosts power against more general alternatives.
Study primitive decompositions for harmonic forms on almost Kähler manifolds.
problem Decomposing harmonic forms on almost Kähler manifolds.
method Proved primitive decompositions for Bott-Chern and Aeppli harmonic forms in specific bidegrees.
result Optimal bidegrees for primitive decompositions of harmonic forms.
New findings show fundamental group is not audible in spherical space forms.
problem Isospectral spherical space forms with non-cyclic fundamental groups.
method Revisited and found new examples of spherical space forms.
result Fundamental group is not audible among spherical space forms.
The paper explores spaces of Kähler and symplectic forms on 4-manifolds.
problem Investigating the properties of Kähler and symplectic forms on 4-manifolds.
method Analyzing the uniqueness, connectedness, and openness of spaces of Kähler forms and introducing holomorphically tamed symplectic forms.
result Formulated a parallel question for holomorphically tamed symplectic forms and related it to Kähler-type symplectic forms.