List of formulas in Riemannian geometry

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This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.

A classical result says that if and only if is locally conformally flat, i. e. if and only if can be covered by smooth coordinate charts relative to which the metric tensor is of the form for some function on the chart.

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations and (but these may not hold at other points in the frame). These coordinates are also called normal coordinates. In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid at the origin of the frame only.

Let be a Riemannian or pseudo Riemanniann metric on a smooth manifold , and a smooth real valued function on Then

is also a Riemannian metric on We say that is (pointwise) conformal to Evidently, conformality of metrics is an equivalence relation. Here are some formulas for conformal changes in tensors associated with the metric. (Quantities marked with a tilde will be associated with , while those unmarked with such will be associated with .)

The "geometer's" sign convention is used for the Hodge Laplacian here. In particular it has the opposite sign on functions as the usual Laplacian.

Let be a smooth manifold and let be a one parameter family of Riemannian or pseudo Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives exist and are themselves as differentiable as necessary for the following expressions to make sense. is a one parameter family of symmetric 2 tensor fields.