In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to:
A connection for which the covariant derivatives of the metric on E vanish. A principal connection on the bundle of orthonormal frames of E. A special case of a metric connection is a Riemannian connection; there exists a unique such connection which is torsion free, the Levi-Civita connection. In this case, the bundle E is the tangent bundle TM of a manifold, and the metric on E is induced by a Riemannian metric on M. Another special case of a metric connection is a Yang–Mills connection, which satisfies the Yang–Mills equations of motion. Most of the machinery of defining a connection and its curvature can be worked through without requiring any compatibility with the bundle metric. However, once one does require compatibility, this metric connection defines an inner product, Hodge star (which additionally needs a choice of orientation), and Laplacian, which are required to formulate the Yang–Mills equations.
Definition Let σ , τ {\displaystyle \sigma ,\tau } be any local sections of the vector bundle E, and let X be a vector field on the base space M of the bundle. Let ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } define a bundle metric, that is, a metric on the vector fibers of E. Then, a connection D on E is a metric connection if:
d ⟨ σ , τ ⟩ = ⟨ D σ , τ ⟩ + ⟨ σ , D τ ⟩ . {\displaystyle d\langle \sigma ,\tau \rangle =\langle D\sigma ,\tau \rangle +\langle \sigma ,D\tau \rangle .}
Here d is the ordinary differential of a scalar function. The covariant derivative can be extended so that it acts as a map on E-valued differential forms on the base space:
D : Γ ( E ) ⊗ Ω p ( M ) → Γ ( E ) ⊗ Ω p + 1 ( M ) . {\displaystyle D:\Gamma (E)\otimes \Omega ^{p}(M)\to \Gamma (E)\otimes \Omega ^{p+1}(M).}
One defines D X f = d X f ≡ X f {\displaystyle D_{X}f=d_{X}f\equiv Xf} for a function f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} , and
D ( σ ⊗ ω ) = D σ ∧ ω + σ ⊗ d ω {\displaystyle D(\sigma \otimes \omega )=D\sigma \wedge \omega +\sigma \otimes d\omega }
where σ ∈ Γ ( E ) {\displaystyle \sigma \in \Gamma (E)} is a local smooth section for the vector bundle and ω ∈ Ω p ( M ) {\displaystyle \omega \in \Omega ^{p}(M)} is a (scalar-valued) p-form. The above definitions also apply to local smooth frames as well as local sections.
Metric versus dual pairing The bundle metric ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } imposed on E should not be confused with the natural pairing ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} of a vector space and its dual, which is intrinsic to any vector bundle. The latter is a function on the bundle of endomorphisms End ( E ) = E ⊗ E ∗ , {\displaystyle {\mbox{End}}(E)=E\otimes E^{*},} so that
( ⋅ , ⋅ ) : E ⊗ E ∗ → M × R {\displaystyle (\cdot ,\cdot ):E\otimes E^{*}\to M\times \mathbb {R} }
pairs vectors with dual vectors (functionals) above each point of M. That is, if { e i } {\displaystyle \{e_{i}\}} is any local coordinate frame on E, then one naturally obtains a dual coordinate frame { e i ∗ } {\displaystyle \{e_{i}^{*}\}} on E* satisfying ( e i , e j ∗ ) = δ i j {\displaystyle (e_{i},e_{j}^{*})=\delta _{ij}} . By contrast, the bundle metric ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is a function on E ⊗ E , {\displaystyle E\otimes E,}
⟨ ⋅ , ⋅ ⟩ : E ⊗ E → M × R {\displaystyle \langle \cdot ,\cdot \rangle :E\otimes E\to M\times \mathbb {R} }
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