Connection on exterior and symmetric powers - Imported Knowlpedia Content

Definition

Let $E\to M$ be a real or complex vector bundle with connection $\nabla$ . The induced connections on the exterior power $\Lambda^kE$ and symmetric power $\operatorname{Sym}^kE$ are the unique connections satisfying

\nabla_X(s_1\wedge\cdots\wedge s_k) = \sum_{i=1}^k s_1\wedge\cdots\wedge\nabla_Xs_i\wedge\cdots\wedge s_k

and the analogous formula with the symmetric product. Equivalently, take the iterated tensor-product connection on $E^{\otimes k}$ ; it preserves alternating and symmetric tensors and therefore descends to both power bundles. Both constructions use the same scalar field as $E$ and require $k\geq0$ .

Parallel transport and curvature

If $T_\gamma:E_{\gamma(0)}\to E_{\gamma(1)}$ is parallel transport for $\nabla$ , the induced transports are $\Lambda^kT_\gamma$ and $\operatorname{Sym}^kT_\gamma$ . This characterization immediately shows that a connection preserving a bundle metric induces metric connections on the corresponding power bundles.

Curvature acts by the derived exterior- or symmetric-power representation. For decomposable exterior tensors,

R^{\Lambda^kE}(X,Y)(s_1\wedge\cdots\wedge s_k) = \sum_{i=1}^k s_1\wedge\cdots\wedge R^E(X,Y)s_i\wedge\cdots\wedge s_k,

and the symmetric formula is identical with $\wedge$ replaced by the symmetric product. In particular, a flat connection induces flat connections on every exterior and symmetric power.

Important special cases

The zeroth powers are the trivial line bundle with its trivial connection, and the first powers recover $(E,\nabla)$ . If $E$ has rank $r$ , the connection on $\Lambda^rE=\det E$ is the determinant connection; in a local frame its connection one-form is the trace of the connection matrix, and its curvature is $\operatorname{tr}(R^E)$ .

For a line bundle $L$ , $\operatorname{Sym}^kL\cong L^{\otimes k}$ , and the induced local connection one-form is $k$ times that of $L$ . A connection chosen independently on $\Lambda^kE$ is a near miss: it need not arise from any connection on $E$ .

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Graduate Texts in Mathematics 176, Springer, 2018. DOI record. Relevant: Chapter 4, connections and induced connections on tensor bundles.
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. Chapter record. Relevant: Chapter II, induced connections and their curvature actions on associated tensor and Clifford modules.