On Symmetric Brackets Induced by Linear Connections

In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian metric structure, we describe the symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. We formulate a generalization of the fundamental theorem of Riemannian geometry. In particular, we obtain an explicit formula of the Levi-Civita connection. We also present some symmetric brackets on almost Hermitian manifolds and discuss the first canonical Hermitian connection. Given a symplectic structure, we describe symplectic connections using symmetric brackets. We define a symmetric bracket of smooth functions on skew-symmetric algebroids with the metric structure and show that it has properties analogous to the Lie bracket of Hamiltonian vector fields on symplectic manifolds.

Minimal submanifolds from the abelian Higgs model

Given a Hermitian line bundle $$L\rightarrow M$$ L → M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as $$\epsilon \rightarrow 0$$ ϵ → 0 , of couples $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) critical for the rescalings $$\begin{aligned} E_\epsilon (u,\nabla )=\int _M\Big (|\nabla u|^2+\epsilon ^2|F_\nabla |^2+\frac{1}{4\epsilon ^2}(1-|u|^2)^2\Big ) \end{aligned}$$ E ϵ ( u , ∇ ) = ∫ M ( | ∇ u | 2 + ϵ 2 | F ∇ | 2 + 1 4 ϵ 2 ( 1 - | u | 2 ) 2 ) of the self-dual Yang–Mills–Higgs energy, where u is a section of L and $$\nabla $$ ∇ is a Hermitian connection on L with curvature $$F_{\nabla }$$ F ∇ . Under the natural assumption $$\limsup _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon )<\infty $$ lim sup ϵ → 0 E ϵ ( u ϵ , ∇ ϵ ) < ∞ , we show that the energy measures converge subsequentially to (the weight measure $$\mu $$ μ of) a stationary integral $$(n-2)$$ ( n - 2 ) -varifold. Also, we show that the $$(n-2)$$ ( n - 2 ) -currents dual to the curvature forms converge subsequentially to $$2\pi \Gamma $$ 2 π Γ , for an integral $$(n-2)$$ ( n - 2 ) -cycle $$\Gamma $$ Γ with $$|\Gamma |\le \mu $$ | Γ | ≤ μ . Finally, we provide a variational construction of nontrivial critical points $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral $$(n-2)$$ ( n - 2 ) -varifolds in an arbitrary closed Riemannian manifold.

ON THE ALGEBRAIC HOLONOMY OF STABLE PRINCIPAL BUNDLES

Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over ℂ. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group to H. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction (which means that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group), then EH is unique in the following sense: For any other minimal reduction of structure group (H′, EH′) of EG to some reductive subgroup H′, there is some element g ∈ G such that H′ = g-1Hg and EH′ = EHg. As an application, we show the following: Let M be a simply connected, irreducible smooth complex projective variety of dimension n such that the Picard number of M is one. If the canonical line bundle KM is ample, then the algebraic holonomy of the holomorphic tangent bundle T1, 0M is GL (n, ℂ). If [Formula: see text] is ample, the rank of the Picard group of M is one, the biholomorphic automorphism group of M is finite, and M admits a Kähler–Einstein metric, then the algebraic holonomy of T1, 0M is GL (n, ℂ). These answer some questions posed in V. Balaji and J. Kollár, Publ. Res. Inst. Math. Sci.44 (2008) 183–211.

Gray identities, canonical connection and integrability

We characterize quasi-Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related to the third Gray identity and in the almost-Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames previously introduced by the second author. By using such frames we also give a simpler and shorter proof of a theorem of Goldberg. Furthermore, we study almost-Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi-Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

SLANT CURVES IN CONTACT PSEUDO-HERMITIAN 3-MANIFOLDS

By using the pseudo-Hermitian connection (or Tanaka–Webster connection) $\widehat \nabla $, we construct the parametric equations of Legendre pseudo-Hermitian circles (whose $\widehat \nabla $-geodesic curvature $\widehat \kappa $ is constant and $\widehat \nabla $-geodesic torsion $\widehat \tau $ is zero) in S3. In fact, it is realized as a Legendre curve satisfying the $\widehat \nabla $-Jacobi equation for the $\widehat \nabla $-geodesic vector field along it.

Onm-accretive Schrödinger-type operators with singular potentials on manifolds of bounded geometry

We consider a Schrödinger-type differential expression∇∗ ∇+V, where∇is aC∞-bounded Hermitian connection on a Hermitian vector bundleEof bounded geometry over a manifold of bounded geometry(M,g)with positiveC∞-bounded measuredμ, andVis a locally integrable linear bundle endomorphism. We define a realization of∇∗ ∇+VinL2(E)and give a sufficient condition for itsm-accretiveness. The proof essentially follows the scheme of T. Kato, but it requires the use of a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of solution to a certain differential equation onM.