In this article, I am going to calculate the Gaussian curvature via connection forms.
Suppose that is a local parametrization on a surface in so that is an orthogonal coordinate system, which means that the tangent vectors and are orthogonal at each point of . Here is an open subset of . The first fundamental form on are given by the following three functions: , , . Here denotes the induced metric on .
Let , . Then forms an orthonormal frame on . Suppose that is the dual coframe to . Then we know , . Let denote the connection one form on . Then
,
.
We obtain that
.
The Gauss-Codazzi equation tells us that
.
Hence
.
As an application, let us calculate the Gaussian curvature when the coordinate system is isothermal. Recall that a coordinate system on a surface is called isothermal if there exists a smooth function so that . Then we can show that
.
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