Supplementary Material on Differentiation of the Volume Form

# Supplementary Material on Differentiation of the Volume Form

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PMATH 465/665: Riemannian Geometry Supplementary Material on Differentiation of the Volume Form Let ( M, g ) be an oriented Riemannian manifold. We use local coordinates x 1 , . . . , x n and denote the Riemannian metric by g ij = g ( ∂x i , ∂x j ). The induced volume form is then μ = p det( g ) dx 1 . . . dx n . We use g ij to denote the inverse matrix to g ij . Let det g denote the determinant of the matrix ( g ij ) of smooth functions. Hence det g is a smooth function on the domain of the coordinate chart. Lemma. Suppose g ij depends smoothly on some parameter t . Then ∂t det g = ∂t g ij g ij det g ∂t μ = 1 2 ∂t g ij g ij μ Proof. The second formula follows easily from the first, using the local coordinate expression for the volume form. Cramer’s rule from linear algebra says g ik G kj = det g δ j i where the matrix ( G ij ) is the adjugate matrix of the matrix ( g ij ). That is, it is the transpose of the matrix of cofactors. Note that
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