arXiv research

Study harmonic mappings and submanifolds using Bochner technique.

New Bochner technique for foliations with non-negative Ricci curvature.

Vanishing theorem on CR manifolds with non-negative curvature.

Einstein's non-symmetric geometry uses Bochner's technique to prove decomposition and vanishing results.

The paper extends Bochner's technique to singular distributions on manifolds.

Study Bochner formula on metric measure spaces for vanishing Betti numbers.

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 …

The main aim of this paper is to extend Bochner's technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge's theory, Bochner-Weitzenbock and Simon's type formulas. Moreover, a few global and local theorem…

Upper bounds on Betti numbers via curvature constraints.

Study on compact hypersurfaces in spheres with Ricci curvature bounds.

Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.

In this short paper, we re-derive the Bochner formula for the Laplacian by considering local variations of volume. The derivation is rooted in the fact that the Laplacian of a function measures the volume variation along the flow of the gradient vector of the function. Possible extensions of this approach/technique are…

We use a modified Bochner technique to derive an inequality relating the nodal set of eigenspinors to eigenvalues of the Dirac operator on closed surfaces. In addition, we apply this technique to solutions of similar spinorial equations.

New proof of Gaffney's inequality for differential forms on manifolds with boundary.

The report presents the theory of harmonic maps from Kähler manifolds.

Extends Bochner's theorem to include small positive Ricci curvature.

Improved estimation of higher order integrals using shrinkage techniques.

By using Bochner technique and gradient estimate, we give the lower bound estimates of the first eigenvalue of Finsler-Laplacian on Finsler manifolds. These results generalize the corresponding famous theorems in the Riemannian geometry.

Proves Bochner's identity on graphs using a new auxiliary graph.

Introduces a universal Bochner formula for scalar curvature.

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two o…

The paper solves linearized Ricci curvature equations on compact manifolds.

Compact Einstein manifolds with specific curvature operators are constant curvature spaces.

We describe a proof of M.T. Anderson's result on the rigidity of complete stationary initial data for the Einstein vacuum equations in spacetime dimension 3 + 1, under an extra assumption on the norm of the stationary Killing vector field. The argument only involves basic comparison geometry along with some Bochner-Wei…

Establishing Hom-versions of Bochner theorems in pseudo-Riemannian Hom-Lie algebras

As an application of the Bochner formula, we prove that if a $2$-dimensional Riemannian manifold admits a non-trivial smooth tangent vector field $X$ then its Gauss curvature is the divergence of a tangent vector field, constructed from $X$, defined on the open subset out the zeroes of $X$. Thanks to the Whitney embedd…

The Bochner tensor is the Kähler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)Kähler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms…

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…

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.

The study improves Bochner inequality on Finsler manifolds to derive important inequalities.

Study resolvents of Bochner Laplacians on compact manifolds.

In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for substatic manifolds. On the other hand, this technique leads to various geometri…

We generalize the classical Bochner formula for the heat flow on evolving manifolds $(M,g_{t})_{t \in [0,T]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $\mathcal{M} = M \times [0,T]$. Our new Bochner formula and the inequalities that follow from it a…

Study of Riemann solitons and $η$-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds.

New rigidity results for tensors on non-compact manifolds with curvature conditions.

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 find necessary and sufficient conditions under which the complex coordinates on a flag manifold of a classical group described in [2] are Bochner coordinates.

We obtain a classification theorem for non Kaehler nearly Kaehler manifolds with vanishing Bochner curvature tensor (introduced by Tricerri and Vanhecke).

The study proves rigidity of Einstein manifolds with specific curvature conditions.

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 generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the re…

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…

Let M^7 a manifold with holonomy in G_2, and Y^3 an associative submanifold with boundary in a coassociative submanifold. In [5], the authors proved that M_{X,Y}, the moduli space of its associative deformations with boundary in the fixed X, has finite virtual dimension. Using Bochner's technique, we give a vanishing t…

The paper derives inequalities and formulas for generalized Ricci flow.

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…

In this paper we find sufficient conditions for the vanishing of the Morse-Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian…

In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain …