Suppose we have a Riemannian metric of the form |d z|^2 /...

Suppose we have a Riemannian metric of the form $|d z|^{2} / h(r)^{2}$ on an open disc of radius $\delta>0$ centred on the origin in $\mathbf{C}$ (possibly all of $\mathbf{C}$ ), where $h(r)>0$ for all $r<\delta$. Show that the curvature $K$ of this metric is given on the punctured disc by the formula $$ K=h h^{\prime \prime}-\left(h^{\prime}\right)^{2}+r^{-1} h h^{\prime} $$

Suppose we have a Riemannian metric of the form $|d z|^{2} / h(r)^{2}$ on an open disc of radius $\delta>0$ centred on the origin in $\mathbf{C}$ (possibly all of $\mathbf{C}$ ), where $h(r)>0$ for all $r<\delta$. Show that the curvature $K$ of this metric is given on the punctured disc by the formula
$$
K=h h^{\prime \prime}-\left(h^{\prime}\right)^{2}+r^{-1} h h^{\prime}
$$

Key Concepts

Radial Coordinates and Differentiation

In problems with radial symmetry, expressing the metric using a radial coordinate allows one to exploit symmetry in the computation of geometric quantities, such as curvature. The derivatives of the radial function in the conformal factor (such as the first and second derivatives) appear in the formula, reflecting how changes in the conformal factor with respect to the radius impact the curvature.

Gaussian Curvature

Gaussian curvature is an intrinsic measure of a surface’s curvature at a point and depends only on the metric. For a given metric, especially in two dimensions, there are explicit formulas relating the curvature to the conformal factor and its derivatives. This curvature quantifies how the surface bends locally and is central in understanding geometric properties of the surface.

Riemannian Metric

A Riemannian metric defines the notion of distance and angles on a smooth manifold by assigning an inner product to the tangent space at every point. It provides the framework for measuring lengths, areas, and curvature, and it is fundamental in the study of differential geometry.

Conformal Metrics

A conformal metric is one that is proportional to a base metric by a positive function, known as the conformal factor. This type of metric preserves angles but not necessarily lengths. The given metric, which has a conformal form, serves as an example where computations simplify due to the relationship between the metric and the underlying Euclidean structure.

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