Study Killing equations for spinor-valued forms on pseudo-Riemannian manifolds.
problem Understanding solutions to Killing equations on pseudo-Riemannian manifolds.
method Introduced a system of partial differential Killing type equations for spinor-valued differential forms.
result Established the relationship between solutions of Killing equations on M and parallel fields on the metric cone over M. Study of Killing spinor-valued forms and their integrability conditions.
problem Understanding Killing spinor-valued forms and their properties.
method Detailed treatment of prolongation and integrability conditions, relating to curvature of the manifold.
result New solutions found that are not from tensor products of Killing spinors and Killing-Yano forms.
Formulas for spectra of higher spin operators on sphere subbundles.
problem Finding spectra of higher spin operators on specific subbundles of spinor-valued tensors.
method Explicit formulas derived for spectra in both even and odd dimensions.
result Spectra formulas for higher spin operators and their squares.
Exterior differential forms with values in the (Kostant's) symplectic spinor bundle on a manifold with a given metaplectic structure are decomposed into invariant subspaces. Projections to these invariant subspaces of a covariant derivative associated to a torsion-free symplectic connection are described.
We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic …
The primary aim of this thesis is to investigate metrics which are induced by a differential form and arise as a critical point of Hitchin's variational principle. Firstly, we investigate metrics associated with the structure group PSU(3) acting in its adjoint representation. We derive various obstructions to the exist…
The underlying even manifold of a super Riemann surface is a Riemann surface with a spinor valued differential form called gravitino. Consequently infinitesimal deformations of super Riemann surfaces are certain infinitesimal deformations of the Riemann surface and the gravitino. Furthermore the action functional of no…
A first-order formulation of gravity is developed in which the fundamental fields consist of an SL(2,C) connection and two spinor-valued 1-forms. It is shown that the first term of an expansion of the Einstein-Hilbert action leads to an action for these fields which consists of dynamic L2 inner products of their covari…
Let λ:G~→G be the non-trivial double covering of the symplectic group G=Sp(V,ω) of the symplectic vector space (V,ω) by the metaplectic group G~=Mp(V,ω). In this case, λ is also a representation of G~ on the vector space V and thus, it gives rise to the representation of $\tilde{G…
We study symplectic manifolds (M2l,ω) equipped with a symplectic torsion-free affine (also called Fedosov) connection ∇ and admitting a metaplectic structure. Let S be the so called symplectic spinor bundle and let RS be the curvature tensor field of the symplectic spinor covariant derivative…
We study conformal Spin-subgeometry of submanifolds in a semi-Riemannian Spin-manifold, focusing on conformal Spin-manifolds (M,[h]) and their Poincaré-Einstein metrics (X,g+). Our approach is based on the spectral theory of Dirac operator in the ambient Spin-manifold, and associated spinor valued meromorp…
Generalizes Gelfand's spectrum to monogenic spinor fields on compact Riemannian manifolds.
problem Extend Gelfand's spectrum concept to non-commutative spaces of spinor fields.
method Use Clifford algebras and monogenic spinor fields, proving essential lemmas and a Stone-Weierstrass theorem.
result Spectrum of monogenic spinor fields on compact Riemannian manifolds is homeomorphic to the manifold itself.
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.
The paper studies local normal forms of singular contact forms and primitive 1-forms.
problem Local normal forms of singular contact forms and primitive 1-forms.
method Combines classical normalization techniques and toric approach.
result Extends and improves previous results on first-order contact forms and primitive 1-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…
Classifies conformal Killing 3-forms on nearly Kähler manifolds.
problem Characterizing conformal Killing forms on nearly Kähler manifolds.
method Fundamental integrability condition for conformal Killing forms.
result All conformal Killing 3-forms are linear combinations of dω and its Hodge dual ∗dω. 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.
New formula found for a unique invariant 8-form on Riemannian manifolds with Spin(9) structure.
problem Finding a new explicit algebraic formula for a unique invariant 8-form.
method Generalizing the standard Kähler 2-form expression, constructing the invariant 8-form from octonion-valued coordinate 1-forms.
result A new explicit algebraic formula for the Spin(9)-invariant 8-form. 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.
Paper presents a new flat triangular form for systems.
problem Creating a structurally flat triangular form for systems.
method Developed a new triangular form based on the extended chained form with conditions for static feedback equivalence.
result Provided conditions for affine input systems to be static feedback equivalent to the new triangular form.
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.
The paper finds Chern-Simons forms for specific classes in simplicial de Rham complex.
problem None explicitly stated; focuses on finding forms.
method Exhibiting Chern-Simons forms of characteristic classes in simplicial de Rham complex.
result Chern-Simons forms for specific characteristic classes identified.
Paper compares two equivariant η-forms, revealing their singular behavior.
problem Comparing two equivariant η-forms to understand their singular behavior.
method Defined and compared equivariant infinitesimal η-form with equivariant η-form modulo exact forms.
result Obtained the singular behavior of the equivariant η-form as a function on the acting Lie group.
Study biharmonic submanifolds in complex and Sasakian space forms.
problem Characterize biharmonic submanifolds in generalized space forms.
method Analyze biharmonicity conditions and derive curvature estimates for various submanifolds.
result Obtain curvature estimates for different types of submanifolds.
Defines a new Poisson bracket on differential forms for symplectic and pseudo-Riemannian metrics.
problem No specific problem stated; defining a new mathematical structure.
method Defined a non-degenerate even Poisson bracket on the algebra of differential forms.
result Established properties and compared with the Koszul-Schouten bracket.
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 …
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.
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.
Researchers found a canonical form for pairs of Hermitian and antilinear operators.
problem Simultaneous normalization of pairs of Hermitian and antilinear operators in differential geometry.
method Finding a canonical form for pairs of Hermitian and antilinear operators.
result Generalized previous results on simultaneous normalization of such pairs.
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.
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.
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 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.
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…
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.