I am kind of confused by the vast number of formulas for computing the Gaussian Curvature. Having a metric tensor / an expression for the line element in 4D (e.g. $t,x,y,z$ or in spherical coordinates $t,r,\theta,\phi$), what is the most general and neat way to calculate the Gaussian Curvature of a metric?
P.S. in order to achieve maximum generality I would consider a metric tensor with off-diagonal terms, so that mixed terms will pop up in the line element expression.
I think you might be talking about the sectional curvature which generalizes the concept of Gaussian curvature to higher dimensions. The sectional curvature $K$ of a 2-dimensional hypersurface is defined as $$K(\mathbf{u},\mathbf{v})={\langle R(\mathbf{u},\mathbf{v})\mathbf{v},\mathbf{u}\rangle\over \langle \mathbf{u},\mathbf{u}\rangle\langle \mathbf{v},\mathbf{v}\rangle-\langle \mathbf{u},\mathbf{v}\rangle^2}$$ where $\mathbf{u}$ and $\mathbf{v}$ are linearly independent vectors tangent to the hypersurface and $R$ is the Riemann curvature tensor. In addition, for a pseudo-Riemannian manifold such as in general relativity, the hypersurface must not be a null hypersurface. The sectional curvature reduces to the standard Gaussian curvature in 3 dimensions. The sectional curvatures in each coordinate direction are related to the the Ricci tensor $R_{ij}$ by the formula: $$R_{ij} v^i v^j = \sum_{\mathbf{e}_i\neq\mathbf{v}} K(\mathbf{e}_i,\mathbf{v})$$ where $\mathbf{v}$ is an arbitrary unit vector at a point and $\mathbf{e}_i$ is a set of $n-1$ ($n$ is the dimension) orthonormal basis vectors that are also orthonormal to $\mathbf{v}$.
