Supplementary Material on Differentiation of the Volume Form

Supplementary Material on Differentiation of the Volume Form

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern