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 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. The expression for the variation of the area functional of the second fundamental form of a hypersurface in a Euclidean space involves the so-called "mean curvature of the second fundamental form". Several new characteristic properties of (hyper)spheres, in which the mean curvature of the second fundamental form occurs…
Study proves higher-order conformal forms don't exist in odd dimensions.
problem Proving non-existence of higher-order conformal forms in odd dimensions.
method Analyzing conformal hypersurface embeddings and differential order invariants.
result General non-existence of higher-order conformal forms in odd dimensions.
The paper defines and calculates fourth fundamental form and i-th curvatures for hypersurfaces in 4D Euclidean space.
problem Calculating curvatures for hypersurfaces in 4D Euclidean space.
method Defining fourth fundamental form and i-th curvatures for hypersurfaces, calculating them on rotational hypersurface, and studying hypersurfaces satisfying a specific differential equation.
result Fourth fundamental form and i-th curvatures are defined and calculated for hypersurfaces in 4D Euclidean space.
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.
Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.
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.
Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.
problem Proving non-degeneracy of critical points for a manifold's squared norm of second fundamental form.
method Generic Riemannian metric and conformal class restriction.
result Squared norm of the second fundamental form is a Morse function with non-degenerate critical points.
In this paper we show that a complete and non-compact surface immersed in the Euclidean space with quadratic extrinsic area growth has finite total curvature provided the surface has tamed second fundamental form and admits total curvature. In such a case we obtain as well a generalized Chern-Osserman inequality. In th…
We give a definition of `coherent tangent bundles', which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisf…
The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.
We obtain an infinite family of complete non embedded rotational surfaces in R3 whose second fundamental forms have length equal to one at any point. Also we prove that a complete rotational surface with second fundamental form of constant length is either a round sphere, a circular cylinder or, up to a homo…
An expression for the first variation of the area functional of the second fundamental form is given for a hypersurface in a semi-Riemannian space. The concept of the "mean curvature of the second fundamental form" is then introduced. Some characterisations of extrinsic hyperspheres in terms of this curvature are given…
We show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN≤κ≤0 are proper, (compact if N is compact). In addition, if N is Hadamard then M has finite topology. We also show that the fundamental tone is an obstruction fo…
New insights into cohomology of closed 1-forms.
problem Understanding twisted cohomology of closed 1-forms.
method Construction of examples and analysis of fundamental group representations.
result Non-trivial twisted cohomology of nowhere-vanishing 1-forms.
Researchers classify special curved spheres in a complex space.
problem Classifying special holomorphic two-spheres in a complex Grassmannian.
method Completely classified noncongruent spheres with constant curvature and second fundamental form.
result Found all homogeneous spheres with constant curvature and second fundamental form.
Study characterizes conformal boundaries of de Sitter spacetimes.
problem Characterize conformal infinity of asymptotically de Sitter spacetimes.
method Derive constraints relating stress-energy tensor to conformal geometric data using higher conformal fundamental forms.
result Constraints on stress-energy tensor relate to conformal geometric data.
The study characterizes geometries of hypersurfaces in warped product and conformal manifolds.
problem Characterizing the geometry of hypersurfaces in warped product and conformal manifolds.
method Using higher fundamental forms and conformal metrics, the study characterizes the geometries of hypersurfaces in warped product and conformal manifolds.
result Higher conformal fundamental forms play a critical role in the characterization of the geometry of hypersurfaces in conformal manifolds.
The study proves properties of self-shrinkers with bounded curvature.
problem Characterizing self-shrinkers with bounded curvature.
method Analyzing properties of self-shrinkers in Rn+1 with bounded second fundamental form. result Proves that if the squared norm of the second fundamental form is bounded, it must be constant.
Study properties of surfaces with nonvanishing third fundamental form.
problem Investigate surfaces with finite III-type properties.
method Analyze Laplace operators and third fundamental form of surfaces in E3.
result Present surfaces of revolution with constant R and nonvanishing Gauss curvature.
In this note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.
Classifies 4-manifolds with specific properties and fundamental group.
problem Classifying 4-manifolds with fundamental group Z and boundary. method Topological and smooth classification methods, including Hermitian forms and 2-handlebody constructions. result Every Hermitian form over Z[t±1] arises as the equivariant intersection form of exotic smooth 4-manifolds. The paper classifies 3D self-expanders with specific properties.
problem Classifying self-expanders with constant curvature components.
method Complete classification of 3D self-expanders with specific curvature conditions.
result Completely classified 3D self-expanders with constant curvature components.
The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one …
New exotic structures found on 4-manifolds with specific groups.
problem Finding exotic smooth structures on 4-manifolds with certain fundamental groups.
method Constructing new smooth structures using definite intersection forms and fundamental groups.
result Infinitely many exotic smooth structures on 4-manifolds with fundamental group Z/2Z.
We prove that if the fundamental 4-form of an almost-quaternionic Hermitian manifold (M, Q, g) of dimension at least eight satisfies the conformal-Killing equation, then (M, Q, g) is quaternionic-Kahler.
We classify Lagrangian submanifolds of complex space forms, whose second fundamental form can be written in a certain way, depending on a real parameter. For some special values of this parameter, the resulting submanifolds are ideal in the sense that they realize equality in an inequality for a Chen's delta-curvature.
The complete local classification and geometric description of n-dimensional submanifolds F with recurrent nonparallel second fundamental form in the spaces of constant curvature M(c) are obtained in this article.
The paper finds conditions for certain hypersurfaces to be totally umbilical.
problem Conditions for constant mean curvature hypersurfaces to be totally umbilical.
method Analyzes the traceless part of the second fundamental form.
result Establishes conditions for complete constant mean curvature hypersurfaces to be totally umbilical.
Study shows the second fundamental form of pseudospherical surfaces is universal and not dependent on specific solutions.
problem Dependence of the second fundamental form in local isometric immersions of pseudospherical surfaces.
method Analysis of third order differential equations and jets of finite order.
result The second fundamental form of pseudospherical surfaces is universal and not dependent on the specific solution.
We unveil the geometric nature of the multiplet of fundamental fermions in the Standard Model of fundamental particles as a noncommutative analogue of de Rham forms on the internal finite quantum space.
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…
We show that an equivariantly embedded Hermitian symmetric space in a projective space, which contains neither a projective space nor a hyperquadric as a component, is characterized by their fundamental forms as a local submanifold of the projective space. Using some invariant-theoretic properties of the fundamental fo…
In this paper, we determine all conformal minimal immersions of 2-spheres in complex Grassmann manifold G(2,N;C) with parallel second fundamental form.
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in W2n−1,2 space. result Proves existence of C1 differential structure from weak immersions with bounded second fundamental forms. The paper proves rigidity for shells in non-Euclidean spaces.
problem Proving rigidity for shells in non-Euclidean spaces.
method Analyzing a stretching plus bending functional of an elastic shell in a Riemannian manifold.
result A sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell.
Researchers classify 3D self-shrinkers in 4D space.
problem Classifying complete 3D self-shrinkers with specific properties in Euclidean space.
method Completely classified 3-dimensional complete self-shrinkers with constant norm of the second fundamental form and constant f3 in R4. result A complete classification of 3D self-shrinkers in Euclidean space R4. A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field η. Several sufficient assumptions on such a surface with non-degenerate η-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…
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.
Study characterizes points on projective surfaces using a cubic form.
problem Characterize points on projective surfaces and their impact on Euler characteristic.
method Define local indices for umbilics and godrons, use fundamental cubic form.
result Formulas relating indices to Euler characteristic determine coexistences of points.
No nontrivial harmonic 1-forms on certain gradient Ricci solitons.
problem Existence of nontrivial harmonic 1-forms on gradient Ricci solitons.
method Proving Liouville theorems for harmonic 1-forms.
result No nontrivial L2-integrable harmonic 1-forms on specified solitons. Study on 4-manifolds with exotic smooth structures and Z_2 fundamental group.
problem Exploring smooth structures on 4-manifolds with specific fundamental groups.
method Construction of irreducible, smooth, oriented, closed, definite 4-manifolds with Z_2 fundamental group and specific Betti numbers.
result Proves existence of infinitely many smooth structures on definite 4-manifolds with positive second Betti number and Z_2 fundamental group.
The study explores smooth structures on specific four-manifolds with cyclic groups, finding many admit infinitely many smooth structures.
problem Exploring smooth structures on four-manifolds with finite cyclic fundamental groups.
method Analyzes topological four-manifolds with odd intersection forms and diverse fundamental groups.
result Many four-manifolds with cyclic fundamental groups admit infinitely many distinct smooth structures.
The paper proves rigidity and vanishing theorems for translating solitons.
problem Understanding the properties of translating solitons in geometry.
method Using Sobolev inequalities and Lq-norms, the paper proves rigidity and vanishing theorems. result Translating solitons are shown to be hypersurfaces under certain conditions.
The paper proves the behavior of the second fundamental form for Kaehler submanifolds in Euclidean space.
problem Classifying non-holomorphic Kaehler submanifolds in Euclidean space with low codimension.
method Analyzing the second fundamental form of submanifolds in Euclidean space.
result The second fundamental form behaves pointwise as expected for low codimensions.
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 classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent b…