Riemannian geometry [SEP] named after
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a "Riemannian metric", i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen,
Gauss–Codazzi equations
In Riemannian geometry, the Gauss–Codazzi–Mainardi equations are fundamental equations in the theory of embedded hypersurfaces in a Euclidean space, and more generally submanifolds of Riemannian manifolds. They also have applications for embedded hypersurfaces of pseudo-Riemannian manifolds.
In the classical differential geometry of surfaces, the Gauss–Codazzi–Mainardi equations consist of a pair of related equations. The first equation, sometimes called the Gauss equation (named after Carl Friedrich Gauss), relates the "intrinsic curvature" (or