Conditions found for flag manifolds to use Bochner coordinates.
problem Finding conditions for Bochner coordinates on flag manifolds.
method Analyzing complex coordinates on flag manifolds of classical groups.
result Necessary and sufficient conditions for Bochner coordinates on flag manifolds.
For the twistor spaces of the Bochner-Kähler manifold M=Hl×Pn, systems of holomorphic coordinates are constructed. As an application of them, an explicit description of the moduli space of relative deformations of fibers of M's twistor space is given.
A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kahler metrics in complex dimension n has real …
Extends Bochner's theorem to include small positive Ricci curvature.
problem Classical Bochner theorem limitations.
method Extensions with small positive Ricci curvature.
result Isometry group is finite for small positive curvature.
Study Bochner formula on metric measure spaces for vanishing Betti numbers.
problem Vanishing Betti numbers on metric measure spaces.
method Introduce weighted curvature conditions.
result Vanishing of all Betti numbers.
Proves Bochner's identity on graphs using a new auxiliary graph.
problem Extending Bochner's identity to graph theory.
method Introduces a complete tangent graph to prove the identity.
result Validates Bochner's identity on graphs.
Derives local forms for Bochner-flat (pseudo-)Kähler metrics.
problem Local description of Bochner-flat metrics.
method Local normal forms for c-projectively equivalent metrics.
result Local description of weakly Bochner-flat metrics.
Introduces a universal Bochner formula for scalar curvature.
problem None explicitly stated; focuses on a new formula.
method Introduces a universal Bochner formula.
result Contains special cases like stability inequality and Schrödinger-Lichnerowicz-type formula.
The study generalizes Bochner Laplacian results to Riemann surfaces.
problem Analyzing curvature vanishing line bundles on Riemann surfaces.
method Exploiting the relation of Bochner Laplacian on tensor powers with sR Laplacian.
result Bergman kernel expansion for semi-positive line bundles.
Generalizes Bochner formula to path space for Ricci flow.
problem Characterize solutions of the Ricci flow.
method Generalizes Bochner formula to parabolic path space.
result Characterizations of the Ricci flow from Bochner inequalities.
Study quantizes eigenstates of Bochner-Laplacian on symplectic manifolds.
problem Quantizing eigenstates of the Bochner-Laplacian on symplectic manifolds.
method Using eigenstates of the renormalized Bochner Laplacian, we apply Berezin-Toeplitz quantization.
result The quantization has correct semiclassical behavior and a corresponding star-product is constructed.
Study harmonic mappings and submanifolds using Bochner technique.
problem Classical theorems in harmonic mappings and submanifolds.
method Generalized Bochner technique.
result New insights into classical theorems.
Establishing Hom-versions of Bochner theorems in pseudo-Riemannian Hom-Lie algebras
problem Killing vectors and Bochner-type theorems in pseudo-Riemannian Hom-Lie algebras
method Hom-versions of Bochner theorems
result Space of Killing vectors forms a totally geodesic Hom-Lie subalgebra
We apply the results from the article Cahen, Schwachhöfer: Special symplectic connections, to the case of Bochner-Kaehler metrics. We obtain a (local) classification of these based on the orbit types of the adjoint action in su(n,1). The relation between Sasaki and Bochner-Kaehler metrics in cone and transveral metri…
Derives Bochner formulas for the Weyl tensor on 4D Einstein manifolds.
problem Understanding the geometry of Einstein manifolds through the Weyl tensor.
method Derives higher-order Bochner type formulas for the Weyl tensor on 4D Einstein manifolds.
result Proves a second Bochner type formula for the Weyl tensor.
New Bochner technique for foliations with non-negative Ricci curvature.
problem Analyzing foliations with non-negative transverse Ricci curvature.
method Generalizing Bochner technique to foliations with non-negative transverse Ricci curvature.
result Obtained a new vanishing theorem for basic cohomology.
We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.
Study eigenvalues of Bochner Laplacian on symplectic manifolds.
problem Understanding low-lying eigenvalues of Bochner Laplacian on symplectic manifolds.
method Analyzes high tensor powers of positive line bundles on symplectic manifolds.
result Asymptotic expansions for low-lying eigenvalues.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
Study resolvents of Bochner Laplacians on compact manifolds.
problem Analyzing the resolvents of Bochner Laplacians in the semiclassical limit.
method Introducing Heisenberg semiclassical pseudodifferential operators to study sections of line bundles.
result Resolvents and spectral projections of Bochner Laplacians are studied in the large power limit.
Einstein's non-symmetric geometry uses Bochner's technique to prove decomposition and vanishing results.
problem Analyzing Einstein's non-symmetric geometry with Bochner's technique.
method Defining concepts, proving decomposition formula, and showing vanishing results.
result Vanishing results about the null space of Bochner and Hodge type Laplacians.
Vanishing theorem on CR manifolds with non-negative curvature.
problem Vanishing theorem for Betti numbers on CR manifolds.
method Application of Bochner technique to CR manifolds with non-negative curvature.
result Proof of vanishing theorem for Betti numbers.
Study of Riemann solitons and η-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds.
problem Analyzing soliton behaviors on Bochner-flat Lorentzian Kähler spacetime manifolds.
method Deriving explicit formulas for soliton parameters and analyzing their behaviors.
result Criteria for shrinking, steady, and expanding behaviors of solitons.
The paper extends Bochner's technique to singular distributions on manifolds.
problem Analyzing the curvature and null space of Hodge Laplacian on singular distributions.
method Defining modified statistical connection, exterior derivative, and Weitzenbock type curvature operator.
result Derivation of Bochner-Weitzenbock type formula leading to vanishing theorems.
B. Y. Chen established sharp inequalities between certain Riemannian invariants and the squared mean curvature for submanifolds in real space form as well as in complex space form. In this paper we generalize Chen inequalities for submanifolds of Bochner Kaehler manifolds. Moreover, we consider CR-warped product subman…
We obtain a classification theorem for non Kaehler nearly Kaehler manifolds with vanishing Bochner curvature tensor (introduced by Tricerri and Vanhecke).
Derives spectral density function for symplectic manifolds.
problem Calculating spectral density functions on symplectic manifolds.
method Explicit local formula derivation for spectral density function.
result Explicit formula for spectral density function.
In this paper, we prove that complete gradient steady Kähler-Ricci solitons with harmonic Bochner tensor are necessarily Kähler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) Kähler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times \mathbb{C}^{n…
We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical…
We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we…
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
We construct a Kahler structure (which we call a generalised Kahler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a 1-parameter family of compatible Sasaki structures. We determine those generalised Kahler cones which are Bochner-flat and we study their local geometry. We prov…
The following result is proved: Consider a 4-dimensional Kaehler manifold M with nonvanishing Bochner tensor B. Then any holomorphic transformation of M, which preserves B is a homothety.
Study on symplectic manifolds finds asymptotic expansions for Bergman kernels.
problem Analyzing asymptotic expansions of Bergman kernels on symplectic manifolds.
method Established asymptotic expansions for renormalized Bochner Laplacians on high tensor powers of positive line bundles.
result Full off-diagonal asymptotic expansion for generalized Bergman kernels.
The main purpose of this article is to prove that there exist no proper AK3-manifold of dimension 2n≥6 with vanishing Tricerri-Vanhecke Bochner curvature tensor and constant scalar curvature.
Study on convergence rates of degenerate SDEs using Fisher information and generalized Bochner's formula.
problem Analysis of dynamical behaviors of degenerate stochastic differential equations.
method Use of Fisher information as Lyapunov functional, generalized Gamma calculus, and generalized Bochner's formula.
result Derivation of convergence rate conditions and examples in specific sub-Riemannian structures.
New proofs of geometric inequalities using Bochner formulas.
problem Geometric inequalities and mixed volumes in convex geometry.
method Reduction to Bochner formulas via spectral theorem.
result New, simpler proofs of Alexandrov-Fenchel and Alexandrov's inequalities.
The study proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(κ) spaces.
problem Analyzing harmonic maps from Riemannian polyhedra into CAT(κ) spaces.
method Computing a target variation formula to derive Liouville-type theorems and Bochner formulas.
result Proves Liouville-type theorems and Bochner formulas for harmonic maps into CAT(1) spaces.
New stability and isolation results for Einstein manifolds.
problem Stability and isolation of Einstein manifolds.
method Conditions on Weyl tensor for AH and ALE manifolds, Bochner tensor for Kähler and Sasaki manifolds.
result Established new stability criteria and isolation results for various types of Einstein manifolds.
It is proved that if an almost Hermitian manifold of dimension greater than 4 has vanishing (classical) Bochner curvature tensor and is not Kaehlerian at a point, then it is flat in a neighbourhood of this point.
New proof using Bochner technique for compact surfaces.
problem Proving a classical result about compact surfaces.
method Application of Bochner formula and Whitney embedding theorem.
result Compact orientable surfaces with no tangent vector field zeroes are diffeomorphic to a torus.
The paper constructs quantizations for symplectic manifolds with specific Laplacian properties.
problem Quantization of compact symplectic manifolds with higher Landau levels.
method Develops Berezin-Toeplitz quantization using a Bochner Laplacian with specific spectral properties.
result The quantization provides a formal star-product for the lowest Landau level.
Study the Bochner-Schrödinger operator's trace in semiclassical limit.
problem Trace formula for Bochner-Schrödinger operator on tensor powers of line and vector bundles.
method Semiclassical analysis of the Bochner-Schrödinger operator Hp on tensor powers of a Hermitian line bundle and vector bundle. result Complete asymptotic expansion of the trace of φ(Hp) in the semiclassical limit po∞. We study curvature properties of four-dimensional almost Hermitian manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke. We give local structure theorems for such Kaehler manifolds, and find out several examples related to the theorems.
The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
problem Asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
method Analyzes the Bochner-Schrödinger operator Hp and its function φ(Hp) in L2(X,Lp⊗E), providing an asymptotic expansion of its smooth Schwartz kernel. result The trace of the operator φ(Hp) admits a complete asymptotic expansion in powers of p−1/2 as po∞. This paper analyzes the Bochner formula for Riemannian flows and derives eigenvalue estimates.
problem Analyzing the Bochner formula for Riemannian flows and deriving eigenvalue estimates.
method The approach involves studying the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M, splitting it into two parts, and establishing eigenvalue estimates.
result Established an eigenvalue estimate of the basic Laplacian on basic forms, and discussed the limiting case of the estimate.
Using as an underlying manifold an alpha-Sasakian manifold we introduce warped product Kaehler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kaehler manifold is of quasi-constant holomorphic sectional curvatures with special distribution. Conversely, we prov…
Study the Bochner-Schrödinger operator on symplectic manifolds, proving gap existence and asymptotic kernel behavior.
problem Analyzing the spectrum and asymptotic behavior of the Bochner-Schrödinger operator on symplectic manifolds.
method Rough asymptotic description, existence proof, off-diagonal exponential estimate, complete asymptotic expansion.
result Existence of gaps in the spectrum and asymptotic kernel behavior.