We are interested in the class, in the Elie Cartan sense, of left invariant forms on a Lie group. We construct the class of Lie algebras provided with a contact form and classify the frobeniusian Lie algebras up to a contraction. We also study forms which are invariant by a subgroup. We show that the simple group SL(2n…
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Survey on invariant conformal Killing forms on Lie groups.
problem Understanding invariant conformal Killing forms on Lie groups.
method Review of recent results and mention of open questions.
result Discussion of recent findings and open research areas.
We study -ary commutative superalgebras and -algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their -ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
Contact forms on SL(2p)math.DG
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.
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.
Study real hypersurfaces in complex space forms for an inequality involving a contact invariant.
problem Understanding real hypersurfaces in complex space forms and their properties.
method Investigating real hypersurfaces that achieve equality in a specific inequality involving a contact invariant.
result Characterized real hypersurfaces in complex space forms achieving the equality in the inequality.
We study the relation between -anti-invariant -forms and pseudoholomorphic curves in this paper. We show the zero set of a closed -anti-invariant -form on an almost complex -manifold supports a -holomorphic subvariety in the canonical class. This confirms a conjecture of Draghici-Li-Zhang. A higher di…
The paper extends the classification of bi-invariant 2-forms to infinite-dimensional Lie groups.
problem Classifying bi-invariant 2-forms on infinite-dimensional Lie groups.
method Generalized the classification from compact Lie groups to arbitrary finite-dimensional Lie groups and then to Milnor regular infinite-dimensional Lie groups.
result The classification of bi-invariant 2-forms extends to all Milnor regular infinite-dimensional Lie groups.
New Arf invariants for colored links determined by linking numbers.
problem Extending Arf invariant to colored links.
method Using generalized Seifert forms to construct quadratic forms and determining Arf invariant.
result New Arf invariants for colored links are determined by linking numbers.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
The object of this paper is to study the invariant submanifolds of Sasakian generalized-Sasakian-space-form. Here, we obtain some equivalent conditions for an invariant submanifold of a Sasakian generalized-Sasakian-space-forms to be totally geodesic.
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
It is well known that there is a unique -invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak -structure. Over the decades, this invariant has been studied extensively and described in several equivalent ways. In the pres…
The study sets constraints on 4-manifold forms linked to specific invariants.
problem Understanding the intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres.
method Analyzes constraints using the μ-bar and κ invariants.
result The difference between κ and -μ-bar for a Seifert rational homology 3-sphere is at most 2, and under certain conditions, it is 0.
Geodesically equivalent Finsler metrics share invariant volume forms and first integrals.
problem Understanding shared properties of geodesically equivalent Finsler metrics.
method Computing first integrals as coefficients of a characteristic polynomial.
result Geodesically invariant functions are first integrals of geodesically equivalent Finsler metrics.
The paper derives curvature inequalities for submersions from quaternionic space forms.
problem Deriving curvature inequalities for submersions from quaternionic space forms.
method Analyzing Ricci and scalar curvatures of horizontal and vertical distributions in anti-invariant submersions.
result Established Ricci curvature inequality for anti-invariant submersions.
The paper defines unimodularity for coisotropic Poisson spaces and discusses invariant volume forms.
problem Understanding unimodularity and invariant volume forms for Hamiltonian dynamics on coisotropic Poisson spaces.
method Introducing multiplicative unimodularity and discussing its properties for coisotropic Poisson homogeneous spaces.
result Existence of invariant volume forms for explicit Hamiltonian systems on coisotropic Poisson spaces.
The paper generalizes CR invariants using renormalized characteristic forms.
problem Defining new CR invariants via renormalized characteristic forms.
method Introducing new curvatures for each renormalized characteristic form.
result The new curvatures' integrals match CR invariants constructed by Marugame.
The paper introduces new invariants for genus one knots and surfaces.
problem Understanding invariants of genus one knots and surfaces.
method Investigating properties of the Alexander form of 3-manifolds to extract invariants of Seifert surfaces.
result Extracted invariants of genus one Seifert surfaces from the Alexander form of their exteriors.
We study left-invariant Killing -forms on simply connected -step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For , we show that every left-invariant Killing -form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-ze…
The paper classifies symplectic forms on R^4 and determines invariants under symplectomorphisms.
problem Classifying symplectic forms on R^4 under symplectomorphisms.
method Using pfaffian and sum function invariants, the paper provides a complete description of orbit spaces and determines global invariants.
result The paper provides a complete classification of symplectic forms on R^4 under symplectomorphisms, providing necessary conditions for intertwining.
The paper provides formulae for CR invariants in Sasakian η-Einstein manifolds.
problem Calculating CR invariants for a specific class of manifolds.
method Using renormalized characteristic forms, the Burns-Epstein invariant and other CR invariants are derived.
result The derived invariants are algebraically independent.
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 …
Twisted Analytic Torsionmath.DG
We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differentia…
The paper derives optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms.
problem Understanding geometric properties of bi-slant submanifolds in metallic Riemannian product space forms.
method Deriving generalized Wintgen inequality, optimal inequalities involving δ-invariants, Ricci curvature, shape operator invariants, and generalized normalized δ-Casorati curvatures.
result Established optimal inequalities for bi-slant submanifolds in metallic Riemannian product space forms.
The paper derives inequalities for Riemannian submersions and applies them to specific space forms.
problem Deriving inequalities for Riemannian submersions.
method Introducing and deriving inequalities for vertical, horizontal, and mixed distributions of Riemannian submersions.
result Established relationships between intrinsic and extrinsic invariants of Riemannian submersions.
Enhances knot invariants using bilinear forms on vector spaces.
problem Improving classical and virtual knot invariants.
method Uses bilinear forms on vector spaces indexed by pairs of elements of a finite quandle.
result New enhanced invariants of knots and links.
Study invariant CKY 2-forms on 5D Lie groups, classifying and determining their properties.
problem Classifying and understanding CKY 2-forms on 5D Lie groups.
method Classification and analysis of 5D metric Lie algebras with CKY tensors.
result First examples of CKY 2-forms on metric Lie algebras without Sasakian structures.
Defines and proves CR invariants on five-manifolds.
problem Defines and studies CR invariants on CR five-manifolds.
method Defines global secondary CR invariants and proves their linear combination.
result Any global secondary CR invariant is a linear combination of total -curvature, total -curvature, and a local CR invariant.
New cubic forms linked to η-invariants and mod 2 indices.
problem Anomaly cancellation and modularity in 12-manifolds.
method Combination of Witten classes and affine character.
result Relates cubic forms to η-invariants and mod 2 indices.
This paper proves a conjecture linking quantum modular forms and WRT invariants for specific graphs.
problem Proving a conjecture about quantum modular forms and WRT invariants for unimodular H-graphs.
method Constructed finite sums of rational functions, studied weighted Gauss sums, and combined results to prove the conjecture.
result WRT invariants of H-graphs yield quantum modular forms of depth two and weight one.
Using our earlier proposal for Ramond-Ramond fields in an H-flux on loop space, we extend the Hori isomorphism of Bouwknegt-Evslin-Mathai from invariant differential forms, to invariant exotic differential forms such that the momentum and winding numbers are exchanged, filling in a gap in the literature. We also extend…
Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2math.DG
Normal forms and invariants for nondegenerate hypersurfaces in C^2.
problem Equivalence problem for nondegenerate real hypersurfaces in C^2.
method Equivariant moving frames and invariant differentiation.
result A single real differential invariant of order 7 generates the entire algebra of differential invariants for nondegenerate real hypersurfaces at singularly umbilic points.
Extended metric defined on Siegel-Jacobi space using invariant forms.
problem Defining a metric on the extended Siegel-Jacobi upper half space.
method Matrix embedding, pre-Iwasawa decomposition, invariant forms, sum of squares of forms.
result Invariant metric on the extended Siegel-Jacobi upper half space is derived.
Certain basic inequalities between intrinsic and extrinsic invariants for a submanifold in a (k, m)-contact space form are obtained. As applications we get some results for invariant submanifolds in a (k,m)-contact space form.
Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.
problem Constructing local models for vector bundles on Kähler manifolds.
method Applying Geometric Invariant Theory to Kähler manifolds to construct analytic GIT-quotients.
result Existence of Weil-Petersson forms on parameter spaces for stable vector bundles.
BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor conn…
The paper explores the connection between Poisson-Lie structures and invariant volume forms in Hamiltonian dynamics.
problem Understanding the relationship between Poisson-Lie structures and invariant volume forms in Hamiltonian systems.
method Analyzing the existence and preservation of invariant volume forms under Hamiltonian vector fields on Poisson-Lie groups.
result A unimodular Poisson-Lie structure ensures the preservation of a multiple of any left-invariant volume, and the existence of a preserving volume form implies unimodularity.
The paper classifies K-contact forms on 3-manifolds and connects their orbits to spectral invariants.
problem Classifying K-contact forms with specific properties on 3-manifolds.
method Analyzing the Reeb vector field and its orbits, proving diffeomorphism results, and relating to spectral invariants.
result Compact 3-manifolds carrying such K-contact forms are diffeomorphic to lens spaces with specific orbit properties.
Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2math.SG
Sharp systolic inequality for invariant tight contact forms on S1-bundles over S2.
problem Finding the shortest period of closed Reeb orbits on invariant tight contact forms.
method Proving a sharp systolic inequality based on the Euler class of the bundle.
result A behavior of the systolic inequality depends on the Euler class of the bundle.
In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980's. We study this invariant from the …
We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order three. We also clarify relations between these invariants.
New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.
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.
Study Dolbeault harmonic forms on Lie group quotients with specific structures.
problem Characterize the space of Dolbeault harmonic (1,1)-forms on compact Lie group quotients.
method Analyze left invariant almost Hermitian structures on 4D Lie groups and their quotients.
result Dimension of Dolbeault harmonic (1,1)-forms depends on existence of a specific anti-self-dual form.
We obtain sharp inequalities involving the Ricci curvature and the scalar curvature for anti-invariant Riemannian submersions from Sasakian space forms onto Riemannian manifolds.
The study of Seifert linking forms for punctured n-manifolds in (2n-1)-space.
problem Understanding Seifert linking forms for punctured n-manifolds.
method Analyzing integer-valued bilinear symmetric forms on and proving their realizability.
result The value modulo two of the invariant equals a specific intersection formula involving Steifel-Whitney class.
We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.