Coordinate transformation law for a Riemannian metric

Statement

Let $(M,g)$ be a Riemannian manifold, and let $G_i$ and $G_j$ be the matrices of $g$ in overlapping coordinate charts $\varphi_i$ and $\varphi_j$ . Put

J_{ij}(x) =D(\varphi_j\circ\varphi_i^{-1})_{\varphi_i(x)},

so that $J_{ij}$ sends $i$ -coordinate components of a tangent vector to $j$ -coordinate components. Then

G_j=J_{ij}^{-T}G_iJ_{ij}^{-1},

or equivalently $G_i=J_{ij}^{T}G_jJ_{ij}$ . This congruence law is exactly the compatibility condition that makes the local matrices represent one global symmetric covariant $2$ -tensor.

Derivation

If a tangent vector has coordinate columns $v_i$ and $v_j=J_{ij}v_i$ , invariance of its squared length gives

v_i^TG_iv_i=v_j^TG_jv_j =v_i^TJ_{ij}^TG_jJ_{ij}v_i.

Since this holds for every $v_i$ , one obtains $G_i=J_{ij}^TG_jJ_{ij}$ .

Relation to the frame bundle

The smooth atlas supplies the $\operatorname{GL}(n,\mathbb R)$ -valued tangent cocycle. A Riemannian metric selects the orthonormal frames, giving an $O(n)$ -reduction of the frame bundle. In orthonormal local frames the metric matrix is $I$ , and transition matrices take values in $O(n)$ .

References

John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018. DOI record. Relevant: Riemannian metrics in local coordinates and orthonormal frames.

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