The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.
problem Understanding the relationship between Chern and Riemann sectional curvatures on Hermitian manifolds.
method Derivation of Chern sectional curvature expressions and subsequent results on Ricci and scalar curvatures.
result A Hermitian metric is Kähler if and only if its Riemann sectional curvature equals its Chern sectional curvature.
New proof shows holomorphic sectional curvature fully determines curvature tensor.
problem Determining the curvature tensor from holomorphic sectional curvature.
method Representation-theoretic means to calculate L2-norm of holomorphic sectional curvature. result Holomorphic sectional curvature fully determines the curvature tensor.
The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.
problem Understanding the ampleness of canonical line bundles for Kähler manifolds with specific curvature properties.
method Introducing a new notion of almost quasi-negative holomorphic sectional curvature and extending the theorem to this setting.
result The theorem is extended to compact Kähler manifolds with almost quasi-negative holomorphic sectional curvature, and a gap-type theorem is derived.
We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature K<c<0 and Ricci curvature Ric>d, where c and d are constants. In particular, a product domain in $\C$ cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe …
The paper proves that Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
problem Diameter rigidity of Kähler manifolds with positive holomorphic sectional curvature.
method Establishing diameter rigidity for Kähler manifolds with positive holomorphic sectional curvature.
result Kähler manifolds with positive holomorphic sectional curvature have a limited diameter.
We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold…
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative c…
It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
Formula for sectional curvature on 2D Lorentzian manifolds derived.
problem Calculating sectional curvature on 2D Lorentzian manifolds.
method Obtained a formula for sectional curvature.
result Formula for sectional curvature on 2D Lorentzian manifolds.
The φ-sectional curvature of statistical structures on almost contact metric manifolds is always non-positive.
problem Analyzing the φ-sectional curvature of statistical structures on almost contact metric manifolds.
method Investigating the φ-sectional curvature induced by a statistical structure and deriving sufficient conditions.
result The φ-sectional curvature is always non-positive.
4-manifolds with nonnegative sectional curvature are area-extremal.
problem Finding extremal properties of 4-manifolds with curvature constraints.
method Analyzing sections in the kernel of a twisted Dirac operator and using the Finsler--Thorpe trick.
result Large classes of compact 4-manifolds are area-extremal.
The paper extends Gray's result to quaternion-Kähler manifolds.
problem Understanding quaternion-Kähler manifolds with non-negative quaternionic sectional curvature.
method Introducing quaternionic sectional curvature, proving Wolf spaces have non-negative curvature, and using nearly Kähler twistor spaces.
result Every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space.
In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming …
Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.
problem Preserving positive sectional curvature in contracting curvature flows in hyperbolic space.
method Homogeneous speed flow with positive sectional curvature, including kth mean curvature flow. result Positive sectional curvature is preserved and the hypersurface contracts to a round point in finite time.
The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.
problem Characterizing compact Kähler manifolds with nonnegative holomorphic sectional curvature.
method Holonomy principle and geometric properties.
result Compact Kähler manifolds with nonnegative holomorphic sectional curvature are projective and rationally connected.
New theorem links quaternionic-Kähler manifolds to symmetric spaces.
problem Understanding curvature properties of quaternionic-Kähler manifolds.
method Analyzing positive scalar curvature and nonnegative sectional curvature.
result Compact quaternionic-Kähler manifolds with these properties are symmetric.
Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant θ-holomorphic sectional curvature for a number θ in (0,π/2) then M is of constant sectional curvature or a Kähler manifold of constant holomorphic sectional curvature. Th…
In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.
Directly proves Wu's theorem on negative curvature metrics.
problem Subadditivity of Hermitian metrics of negative holomorphic sectional curvature.
method Quick direct proof
result Subadditivity of Hermitian metrics of negative holomorphic sectional curvature.
Ricci flow preserves positive sectional curvature on homogeneous spheres
problem Classification of positively curved metrics on homogeneous spaces
method Proving Ricci flow preserves positive sectional curvature on homogeneous spheres
result Completes classification of positively curved metrics on homogeneous spaces
No conformal product structures on compact manifolds with constant curvature.
problem Existence of conformal product structures on compact manifolds with constant curvature.
method Analyzing non-flat manifolds and irreducible, compact locally symmetric spaces of non-positive curvature.
result Compact non-flat manifolds with constant sectional curvature admit no conformal product structure.
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field…
The study characterizes spacetimes with quasi-constant sectional curvature and explores their properties in F(R)-gravity.
problem Characterizing spacetimes with quasi-constant sectional curvature.
method Investigation through examples, proofs, and analysis of energy conditions.
result A spacetime of quasi-constant sectional curvature can represent a Robertson Walker spacetime or a static spacetime.
We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to geometrically realize these results to produce a hypersurface with prescribed sectional curv…
Nonnegative sectional curvature linked to matrix displacement convexity.
problem Nonnegative sectional curvature in Riemannian manifolds.
method Matrix displacement convexity as a criterion for nonnegative sectional curvature.
result Entropy functional matrix displacement convexity implies nonnegative sectional curvature.
Classifies Kähler metrics with constant holomorphic curvature.
problem Classifying Kähler metrics with constant holomorphic sectional curvature.
method Exploiting the geometry of the bundle of 1-jets of holomorphic functions.
result Local classification of Kähler metrics with constant holomorphic sectional curvature.
Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.
problem Rigidity of harmonic maps between manifolds with curvature constraints.
method Proves an extension of Eells-Sampson theorem under positive sectional curvature upper bounds.
result Recover Hamilton's rigidity result for positive Ricci curvature.
In [11], I. M. Gelfand, V. Retakh, and M. Shubin defined the symplectic sectional curvature of a torsion-free connection preserving a symplectic form. The present article defines the corresponding notion of constant symplectic sectional curvature and characterizes this notion in terms of the curvature tensor of the sym…
Every biharmonic Wintgen ideal submanifold in a Riemannian manifold of constant sectional curvature is either minimal or has constant mean curvature.
problem Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
method Show that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of nonpositive constant sectional curvature is minimal and that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature.
result Partial affirmative answers to Chen's conjecture, generalized Chen's conjecture in hyperbolic spaces, and Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.
Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.
problem Finding upper bounds for dimensions of subspaces where holomorphic sectional curvature vanishes.
method Connection with D'Angelo's work on complex subvarieties of real algebraic varieties and decomposition of polynomials into differences of squares.
result An upper bound for the dimensions of these subspaces is found.
New curvature obstruction for Killing vector fields on Lorentzian manifolds.
problem Existence of timelike or causal Killing vector fields on Lorentzian manifolds.
method New curvature obstruction in terms of timelike or null sectional curvature.
result Extension of Gauss-Bonnet-Chern obstruction to non-zero timelike sectional curvature.
The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.
problem Improving inequalities for Kähler-Einstein manifolds.
method Using invariant theory and curvature conditions to express and improve inequalities.
result Improved inequalities for Kähler-Einstein manifolds with smaller pinching constants.
The paper classifies hypersurfaces with constant curvature in Euclidean spaces.
problem Classifying separable hypersurfaces with constant sectional curvature.
method Analytical proof and classification of hypersurfaces in Euclidean spaces.
result Hyperspheres are the only separable hypersurfaces with nonzero constant sectional curvature.
Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
problem Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
method Use Bézout estimates and a Lipschitz weight with finite Monge-Ampère mass
result Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
Study curvature of blown-up manifold, finds negative holomorphic sectional curvature.
problem Proving positive curvature of blowups of manifolds with positive holomorphic sectional curvature.
method Calculates curvature tensor of a specific metric on the blowup's exceptional divisor.
result Holomorphic sectional curvature is negative in some directions for small enough t.
Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
problem Understanding the collapsibility of 2-complexes with specific curvature properties.
method Analyzing the fundamental groups and sectional curvatures of 2-complexes.
result Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant δ depending on the ratio such that $\cosα\ge…
New findings on curvature and null spaces of Laplacians.
problem Relationship between sectional curvature and Laplacian null spaces.
method Analysis of curvature operators and Laplacians on Riemannian manifolds.
result Curvature operator's positivity implies sectional curvature positivity.
We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hyper…
We obtain a locally symmetric Kaehler Einstein structure on the cotangent bundle of a Riemannian manifold of negative constant sectional curvature. Similar results are obtained on a tube around zero section in the cotangent bundle, in the case of a Riemannian manifold of positive constant sectional curvature. The obtai…
We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, i…
We generalize a construction of Hitchin to prove that, given any compact Kähler manifold M with positive holomorphic sectional curvature and any holomorphic vector bundle E over M, the projectivized vector bundle P(E) admits a Kähler metric with positive holomorphic sectional curvature.
If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.
We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature…
The paper classifies translation surfaces with constant curvature in a specific connection.
problem Classifying translation surfaces with constant curvature in a semi-symmetric non-metric connection.
method Completely classified translation surfaces of constant sectional curvature in a semi-symmetric non-metric connection.
result Translation surfaces of constant curvature are generalized cylinders, similar to the Levi-Civita connection but with additional non-constant curvature cases.
Paper studies fundamental groups of certain Ricci solitons.
problem Understanding fundamental groups of specific Ricci solitons.
method Analyzes properties of complete steady gradient Ricci solitons with nonnegative sectional curvature.
result Fundamental groups of these solitons are either trivial or infinite.
Ricci flow deforms metrics with positive curvature to include negative curvature.
problem Preserving positive sectional curvature under Ricci flow in dimension four.
method Evolved cohomogeneity one metrics on S4 and CP2 via Ricci flow. result Metrics with positive sectional curvature lose this property under Ricci flow.