Submanifold in the context of Normal component

In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by ${\displaystyle \mathrm {I\!I} }$ (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.

Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largelyto the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck, Clifford Taubes, Shing-Tung Yau, Richard Schoen, and Richard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to the Poincaré conjecture by Grigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.

In Riemannian geometry, the geodesic curvature ${\displaystyle k_{g}}$ of a curve ${\displaystyle \gamma }$ measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's tangent plane. More generally, in a given manifold ${\displaystyle {\bar {M}}}$, the geodesic curvature is just the usual curvature of ${\displaystyle \gamma }$ (see below). However, when the curve ${\displaystyle \gamma }$ is restricted to lie on a submanifold ${\displaystyle M}$ of ${\displaystyle {\bar {M}}}$ (e.g. for curves on surfaces), geodesic curvature refers to the curvature of ${\displaystyle \gamma }$ in ${\displaystyle M}$ and it is different in general from the curvature of ${\displaystyle \gamma }$ in the ambient manifold ${\displaystyle {\bar {M}}}$. The (ambient) curvature ${\displaystyle k}$ of ${\displaystyle \gamma }$ depends on two factors: the curvature of the submanifold ${\displaystyle M}$ in the direction of ${\displaystyle \gamma }$ (the normal curvature ${\displaystyle k_{n}}$), which depends only on the direction of the curve, and the curvature of ${\displaystyle \gamma }$ seen in ${\displaystyle M}$ (the geodesic curvature ${\displaystyle k_{g}}$), which is a second order quantity. The relation between these is ${\displaystyle k={\sqrt {k_{g}^{2}+k_{n}^{2}}}}$. In particular geodesics on ${\displaystyle M}$ have zero geodesic curvature (they are "straight"), so that ${\displaystyle k=k_{n}}$, which explains why they appear to be curved in ambient space whenever the submanifold is.

In topology, a branch of mathematics, local flatness is a smoothness condition that can be imposed on topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Violations of local flatness describe ridge networks and crumpled structures, with applications to materials processing and mechanical engineering.