# Calculating Gaussian Curvature in 4D

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.

• What is your definition of Gaussian curvature in more than two dimensions? Mar 17, 2022 at 18:07
• @gangio, Gaussian curvature is an intrinsic property of two-dimensional surfaces only. Do you mean perhaps Riemannian curvature tensor?
– JanG
Mar 18, 2022 at 9:01

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 $$\mathbf{v}$$.