On the curvature of tensor product connections and covariant differentials. (English) Zbl 1051.53017

Slovák, Jan (ed.) et al., The proceedings of the 23th winter school “Geometry and physics”, Srní, Czech Republic, January 18–25, 2003. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 72, 135-143 (2004).
The concept of tensor product connection \(K\otimes K'\) of two linear connections \(K\), \(K'\) on vector bundles \(E\), \(E'\) on a given manifold \(M\) was introduced in [L. Mangiarotti and M. Modugno, Connections and differential calculus on fibered manifolds. 3rd ed. of the preprint (University, Florence) (1989; Zbl 0841.53023)].
For any positive integers \(p\), \(q\), \(r\), \(s\) the linear tensor product connection \(k^p_q\otimes k^{\prime r}_s\) is defined as \(\otimes^p K\otimes\otimes^q K^*\otimes\otimes^r K'\otimes\otimes^s K^{\prime *}\), \(K^*\), \(K^{\prime *}\) denoting the dual connections of \(K\), \(K'\), respectively.
This paper consists a detailed investigation of the curvatures \(R[K\otimes K']\), \(R[K^p_q\otimes K^{\prime r}_s]\) of the tensor product connections \(K\otimes K'\), \(K^p_q\otimes K^{\prime r}_s\). Coordinate formulas explain the dependence of \(R[K\otimes K']\), \(R[K^p_q\otimes K^{\prime r}_s]\) on the curvatures \(R[K]\), \(R[K']\).
The authors call a linear symmetric connection \(\Gamma\) on the tangent bundle \(TM\) classical and denote by \(E^{p,r}_{q,s}\) the vector bundle \(\otimes^p E\otimes_M\otimes^q E^*\otimes_M \otimes^r TM\otimes_M\otimes^s T^*M\). Given a connection \(K\) on \(E\) and a classical connection \(\Gamma\), for any section \(\Phi\) of \(E^{p,r}_{q,s}\), the authors state a coordinate formula for the covariant differential of \(\Phi\) with respect to \(K^p_q\otimes\Gamma^r_s\). As particular cases, one obtains the generalized Bianchi identity and a formula which links the second-order covariant differential of any section \(\Phi\) of \(E^{p,r}_{q,s}\) to the curvature \(R[\Gamma^p_q\otimes K^r_s]\).
For the entire collection see [Zbl 1034.53002].


53C05 Connections (general theory)


Zbl 0841.53023