[Paper Review] Connections in holomorphic Lie algebroids

Alexandru D. Ionescu, Gheorghe Munteanu|arXiv (Cornell University)|May 26, 2016

Advanced Differential Geometry Research8 references3 citations

TL;DR

This paper investigates connections in holomorphic Lie algebroids by analyzing the tangent bundle $T^{ inyullet}E$ and its prolongation $\mathcal{T}^{ iny\bullet}E$, introducing a canonical spray via a complex Lagrangian and deriving a nonlinear connection on $T^{ iny\bullet}M$ from the Chern-Lagrange connection on $E$. The key contribution is the construction of an induced nonlinear connection on $T^{ iny\bullet}M$ from a Lagrangian structure on $E$, with explicit formulas for the connection coefficients and adapted frames.

ABSTRACT

The main purpose of this note is the study of the total space of a holomorphic Lie algebroid $E$. The paper is structured in three parts. In the first section we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section is a little broader and includes two approaches to study the geometry of complex manifold $E.$ The first part contains the study of the tangent bundle $T_{C}E=T^{\prime }E\oplus T^{\prime \prime }E$ and its link, via tangent anchor map, with the complexified tangent bundle $% T_{C}(T^{\prime }M)=T^{\prime }(T^{\prime }M)\oplus T^{\prime \prime }(T^{\prime }M).$ A holomorphic Lie algebroid structure has been emphasized on $T^{\prime }E.$ A special study is made for integral curves of a spray on $% T^{\prime }E.$ Theorem 2.1 gives the coefficients of a spray, called canonical, according to a complex Lagrangian on $T^{\prime }E.$ In the second part of section two we study the prolongation $\mathcal{T}% ^{\prime }E$ of $E imes T^{\prime }E$ algebroid structure. In the third section we study how a complex Lagrange (Finsler) structure on $T^{\prime }M$ induces a Lagrangian structure on $E.$ Three particular cases are analyzed by the rank of anchor map, the dimensions of manifold $M$ and the fibre dimension. We obtain the correspondent on $E$ of the well-known (\cite{Mu}) Chern-Lagrange nonlinear connection from $T^{\prime }M$.

Motivation & Objective

• To generalize real Lie algebroid geometry to the holomorphic setting, particularly focusing on tangent and prolongation structures.
• To define and characterize a canonical spray on $T^{ iny\bullet}E$ using a complex Lagrangian, extending variational principles to holomorphic algebroids.
• To establish a correspondence between nonlinear connections on $E$ and those on $T^{ iny\bullet}M$ via the anchor map and induced coordinates.
• To investigate how a Lagrangian structure on $T^{ iny\bullet}M$ induces a Lagrangian and nonlinear connection on the prolongation $\mathcal{T}^{ iny\bullet}E$.
• To prove that the Chern-Lagrange nonlinear connection on $T^{ iny\bullet}M$ arises as the induced connection from a Lagrangian on $E$.

Proposed method

• Introduce holomorphic Lie algebroids via a holomorphic anchor map $\rho: E \to T^\prime M$ and define the Lie bracket on $\Gamma(E)$ using the pullback of the holomorphic bracket on $T^\prime M$.
• Define the tangent bundle $T^\prime E$ as a holomorphic Lie algebroid and construct a complex linear connection adapted to a nonlinear connection on $T^\prime E$.
• Derive the coefficients of a canonical spray on $T^\prime E$ using the variational principle from a regular complex Lagrangian $L^\ast$ via the formula $N_h^k(z,\eta) = \rho_\alpha^k N_h^\alpha(z,u) - u^\alpha \partial \rho_\alpha^k / \partial z^h$.
• Construct the prolongation $\mathcal{T}^\prime E$ of $E$ as a holomorphic algebroid and define an almost tangent structure and Liouville tensor to induce a nonlinear connection.
• Use the complete lift of the anchor map to relate the geometry of $E$ to that of $T^\prime M$, and define adapted frames and cobases via $\delta^\ast \eta^k = \rho_\alpha^k \delta u^\alpha$.
• Establish the equivalence between the nonlinear connection on $T^\prime E$ and the induced connection on $T^\prime M$ via the coordinate transformation $\eta^k = \rho_\alpha^k u^\alpha$.

Experimental results

Research questions

• RQ1 How can the geometry of a holomorphic Lie algebroid $E$ be linearized using a nonlinear connection and a complex Lagrangian?
• RQ2 What is the explicit form of the canonical spray on $T^\prime E$ derived from a regular complex Lagrangian?
• RQ3 How is a nonlinear connection on $T^\prime M$ induced from a nonlinear connection on $E$ via the anchor map?
• RQ4 Can the Chern-Lagrange nonlinear connection on $T^\prime M$ be obtained as the induced connection from a Lagrangian on $E$?
• RQ5 What is the relationship between the adapted frames on $T^\prime E$ and the induced frames on $T^\prime M$?

Key findings

• The canonical spray on $T^\prime E$ is fully determined by the complex Lagrangian $L^\ast$, with coefficients given by $N_h^k(z,\eta) = \rho_\alpha^k N_h^\alpha(z,u) - u^\alpha \partial \rho_\alpha^k / \partial z^h$.
• The induced nonlinear connection on $T^\prime M$ satisfies $\delta^\ast \eta^k = \rho_\alpha^k \delta u^\alpha$, ensuring compatibility between the geometry of $E$ and $T^\prime M$.
• The adapted frame on $T^\prime M$ is given by $\frac{\delta}{\delta z^k} = \frac{\delta^\ast}{\delta z^k}$ and $\frac{\partial}{\partial \eta^k} = \frac{\partial^\ast}{\partial v^k}$, with $v^k = \rho_\alpha^k u^\alpha$.
• The metric tensor $g_{i\bar{j}}(z,\eta)$ on $T^\prime M$ is the pullback of the metric on $E$, expressed as $g_{i\bar{j}}(z,\eta) = \left(g_{i\bar{j}}(z,u)\right)^\ast = \frac{\partial^2 L^\ast}{\partial \eta^i \partial \bar{\eta}^j}$.
• The Chern-Lagrange nonlinear connection on $T^\prime M$ is induced from the Chern-Lagrange connection on $E$ via the anchor map, as shown in Proposition 3.6.
• The nonlinear connection on $\mathcal{T}^\prime E$ is derived from a spray on $\mathcal{T}^\prime E$ using Theorem 2.2, and the induced connection on $T^\prime M$ matches the classical Chern-Lagrange connection.