Study of convex hypersurfaces with specific curvature properties.
problem Characterizing convex hypersurfaces with vanishing Weyl curvature and semi-parallel cubic form.
method Analyzing locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric.
result Classification of such hypersurfaces, excluding flat affine metric cases.
Paper classifies special Euclidean hypersurfaces with specific geometric properties.
problem Classifying Euclidean hypersurfaces with semi-parallel Moebius second fundamental form.
method Complete classification of hypersurfaces with three distinct principal curvatures.
result Classification of Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form.
This paper classifies flat submanifolds with a special type of curvature form.
problem Classifying flat submanifolds with a specific curvature property.
method Using Moebius geometry and curvature operators to classify submanifolds.
result Classification of umbilic-free isometric immersions with flat normal bundle and semi-parallel Moebius second fundamental form.
We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in [Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential Geom. Appl.] stating that every Lagrangian pseudo-parallel submanifold of a complex space form of dimension at least 3 is semi-parallel.
In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians G2(Cm+2). By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in G2(Cm+2).
We classify pseudo parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one. With this result, the non-existence of recurrent as well as semi parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one can also be obtained.
We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.
In the present article we study a special class of surfaces in the four-dimensional Euclidean space, which are one-parameter systems of meridians of the standard rotational hypersurface. They are called meridian surfaces. We classified semi-parallel meridian surface in 4-dimensional Euclidean space E4.
We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.
Formula for spacelike submanifolds in warped products.
problem Finding formulas for spacelike submanifolds in semi-Riemannian warped products.
method Extending Simons' type formulas for spacelike submanifolds in semi-Riemannian warped products.
result Compact spacelike hypersurfaces with parallel mean curvature and non-negative sectional curvature are isoparametric hypersurfaces.
This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose ∗-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel ∗-Ricci tensor in case…
It is proved the non-existence of Hopf hypersurfaces in G2(Cm+2), m≥3, whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle D or its orthogonal …
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.
Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.
problem Characterizing curvature symmetries in lightlike hypersurfaces of indefinite Sasakian manifolds.
method Analyzing properties of lightlike hypersurfaces in indefinite Sasakian manifolds.
result Lightlike hypersurfaces are not locally symmetric, semi-symmetric, or semi-parallel.
The paper classifies λ-biharmonic hypersurfaces in product spaces.
problem Characterizing λ-biharmonic hypersurfaces in product spaces.
method Analyzing λ-biharmonic hypersurfaces in L^{m} imes\mathbb{R} and L^{m}(c) imes\mathbb{R} with various constraints.
result λ-biharmonic hypersurfaces in L^{m} imes\mathbb{R} are either minimal or vertical cylinders.
The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassm…
In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in Sn×R and Hn×R are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of …
In this paper, we study biharmonic hypersurfaces in a product of an Einstein space and a real line. We prove that a biharmonic hypersurface with constant mean curvature in such a product is either minimal or a vertical cylinder generalizing a result of \cite{OW} and \cite{FOR}. We derived the biharmonic equation for hy…
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.
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.
The paper proves unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.
problem Proving unboundedness of Donaldson-Hitchin functionals on G2 and tG2 forms.
method Explicit local calculations combined with covering arguments.
result Proves unboundedness above and below of the Donaldson-Hitchin functionals on G2 and tG2 forms.