[Paper Review] On the curvature of tensor product connections and covariant differentials

Josef Janyška|ArXiv.org|Apr 3, 2003

Advanced Differential Geometry Research1 references11 citations

TL;DR

This paper derives the coordinate formula and geometric description of the curvature of the tensor product connection of two linear connections on vector bundles over the same base manifold. It introduces a generalized covariant differential for tensor fields using a linear connection on a vector bundle and a symmetric linear connection on the base manifold, proving the generalized Bianchi identity and showing that the antisymmetrization of the second-order covariant differential is expressible via the curvature tensors of both connections.

ABSTRACT

We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a linear symmetric connection on the base manifold. We prove the generalized Bianchi identity for linear connections and we prove that the antisymmetrization of the second order covariant differential is expressed via the curvature tensors of both connections.

Motivation & Objective

• To provide a coordinate formula and geometric description of the curvature of the tensor product connection of two linear connections on vector bundles over the same base manifold.
• To define a covariant differential for geometric fields—sections of tensor products of a vector bundle, its dual, and tangent/cotangent bundles—using a linear connection on the vector bundle and a symmetric linear connection on the base manifold.
• To generalize the classical Bianchi identity to the setting of linear connections and prove that the antisymmetrization of the second-order covariant differential is determined by the curvature tensors of both connections.
• To establish a precise relationship between the curvature of the tensor product connection and the curvatures of the individual connections.

Proposed method

• Uses the Froelicher-Nijenhuis bracket to define the curvature of a linear connection as a vertical 2-form with values in the vertical bundle.
• Expresses the curvature of the tensor product connection $K \otimes K'$ in terms of the curvatures of $K$ and $K'$, using coordinate expressions involving connection coefficients $K_j^i{}_{\lambda}$ and their derivatives.
• Introduces a mixed covariant differential $\nabla = \nabla^{(K,\Gamma)}$ combining a linear connection $K$ on a vector bundle and a symmetric linear connection $\Gamma$ on the base manifold.
• Applies the generalized Bianchi identity $[K, R[K]] = 0$ as a foundational tool to derive identities for higher-order covariant derivatives.
• Uses coordinate expressions for the second-order covariant differential $\nabla^2\Phi$ of tensor fields $\Phi$ of type $(p,r)_{q,s}$, involving both $K$ and $\Gamma$ connections.
• Applies antisymmetrization $\operatorname{Alt}$ to the second-order covariant differential and derives its expression in terms of curvature tensors of $K$ and $\Gamma$.

Experimental results

Research questions

• RQ1 How is the curvature of the tensor product connection $K \otimes K'$ expressed in coordinates and geometrically in terms of the curvatures of $K$ and $K'$?
• RQ2 What is the structure of the covariant differential for tensor fields formed from a vector bundle, its dual, and the tangent/cotangent bundles when equipped with a linear connection on the bundle and a symmetric connection on the base manifold?
• RQ3 Does the generalized Bianchi identity hold for the covariant differential of the curvature tensor of a linear connection when combined with a symmetric connection on the base?
• RQ4 How is the antisymmetrization of the second-order covariant differential related to the curvature tensors of the underlying connections?

Key findings

• The curvature of the tensor product connection $K \otimes K'$ is fully determined by the curvatures of $K$ and $K'$, with the coordinate expression given by $R[K \otimes K']_{j}{}^{i}{}_{\lambda\mu} = R[K]_{j}{}^{i}{}_{\lambda\mu} + R[K']_{j}{}^{i}{}_{\lambda\mu}$.
• The generalized Bianchi identity holds: $R[K]_{j}{}^{i}{}_{\lambda\mu;\nu} + R[K]_{j}{}^{i}{}_{\mu\nu;\lambda} + R[K]_{j}{}^{i}{}_{\nu\lambda;\mu} = 0$, where the semicolon denotes covariant derivative with respect to the connection $\nabla^{(K,\Gamma)}$.
• The antisymmetrization of the second-order covariant differential of a tensor field $\Phi$ satisfies $\operatorname{Alt}\nabla^2\Phi = -\frac{1}{2} R[\Gamma^p_q \otimes K^r_s] \circ \Phi$, expressing the result in terms of the curvature of the tensor product connection.
• For a section $\Phi = \phi^i \mathbf{b}_i$, the antisymmetrized second covariant derivative is $\operatorname{Alt}\nabla^2\Phi = -\frac{1}{2} R[K]_j{}^i{}_{\lambda\mu} \phi^j \mathbf{b}_i \otimes d^\lambda \wedge d^\mu$.
• The second covariant derivative of the curvature tensor $R[K]$ satisfies $\operatorname{Alt}\nabla^2 R[K] = -\frac{1}{2} \big( R[K]_p{}^i{}_{\nu_1\nu_2} R[K]_j{}^p{}_{\lambda\mu} - R[K]_j{}^p{}_{\nu_1\nu_2} R[K]_p{}^i{}_{\lambda\mu} - R[\Gamma]_\lambda{}^\omega{}_{\nu_1\nu_2} R[K]_j{}^i{}_{\omega\mu} - R[\Gamma]_\mu{}^\omega{}_{\nu_1\nu_2} R[K]_j{}^i{}_{\lambda\omega} \big) \mathbf{b}^j \otimes \mathbf{b}_i \otimes d^\lambda \wedge d^\mu \otimes d^{\nu_1} \wedge d^{\nu_2}$.
• When $p = q = 0$, the covariant differential $\nabla\Phi$ reduces to the standard covariant derivative of a $(r,s)$-tensor field with respect to the classical symmetric connection $\Gamma$, confirming consistency with classical differential geometry.