Geometry Source for a coordinate-based formula for Gauss from Riemann

[The Bill]

I need a citeable source that gives the formula for the Gaussian curvature at a single point of an intrinsically defined Riemannian or Semi-Riemannian manifold given the intrinsic metric tensor and/or Riemann tensor.

I've got sources for this already, but I'm not "allowed" to use them for this, I need a professionally published journal/textbook source for a citation.

 

[jedishrfu]

Not sure if this what you’re looking for

https://projecteuclid.org/euclid.jdg/1214429996

 

[The Bill]

First, I'm not familiar with the gamma-curly braces function the author is applying to the line element. And second, any formula that uses the line element isn't relevant to this thread. I suppose I didn't make it clear, but I'm looking for any variant of a formula in traditional debauch-of-indices Ricci-style. That's what I meant by coordinate-based in the title. Not using coordinate-free notation or methods at all.

This is for a computational approach, too be read by people who just want to see the math they need to write an algorithm to crunch numbers with, nothing more. The manifold will be defined with specific choices of charts already there for the use of the algorithm, so coordinate-free methods are unnecessary.