Knowlpedia

Definition

Tensor product connection

The connection on a tensor product bundle obtained by differentiating each factor.

Definition

Let E→ME\to M and F→MF\to M be smooth or with ∇E\nabla^E and ∇F\nabla^F. The tensor product connection is the unique connection ∇E⊗F\nabla^{E\otimes F} on the satisfying

∇XE⊗F(s⊗t)=(∇XEs)⊗t+s⊗(∇XFt)\nabla^{E\otimes F}_X(s\otimes t) = (\nabla^E_Xs)\otimes t+s\otimes(\nabla^F_Xt)

for every XX and ss of EE and tt of FF. Equivalently,

∇E⊗F(s⊗t)=∇Es⊗t+s⊗∇Ft.\nabla^{E\otimes F}(s\otimes t) = \nabla^Es\otimes t+s\otimes\nabla^Ft.
Why the formula is well defined

Sections of E⊗FE\otimes F are locally finite sums of decomposable sections s⊗ts\otimes t. The displayed rule is balanced over smooth functions: replacing s⊗fts\otimes ft by fs⊗tfs\otimes t gives the same result because the two occurrences of df\mathrm df agree. It therefore descends from pairs of sections to their tensor product and obeys the Leibniz rule required of a connection.

In local frames, if

∇E=d+Aand∇F=d+B,\nabla^E=\mathrm d+A \qquad\text{and}\qquad \nabla^F=\mathrm d+B,

then the product connection has connection matrix

A⊗IF+IE⊗B.A\otimes I_F+I_E\otimes B.

This expression explains why both factors are differentiated.

Curvature

The contains no mixed term:

RE⊗F=RE⊗IF+IE⊗RF.R^{E\otimes F} = R^E\otimes I_F+I_E\otimes R^F.

Consequently, the tensor product of two flat connections is flat. The converse need not hold, since scalar curvature contributions from the two factors can cancel.

Related induced connections

The same Leibniz principle defines connections on tensor powers, exterior powers, symmetric powers, dual bundles, and bundles of homomorphisms. For example, the is characterized by

X(λ(s))=(∇Xλ)(s)+λ(∇Xs),X\bigl(\lambda(s)\bigr) = (\nabla_X\lambda)(s)+\lambda(\nabla_Xs),

and the connection on Hom⁡(E,F)\operatorname{Hom}(E,F) satisfies

(∇XT)(s)=∇XF(Ts)−T(∇XEs).(\nabla_XT)(s)=\nabla^F_X(Ts)-T(\nabla^E_Xs).
References
  1. John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. DOI record. Relevant: Chapter 5, connections and induced connections on tensor bundles.