Curvature of a curve

Curvatures of curves on surfaces

• Normal curvature
• Geodesic curvature

Curvature of Surfaces

Introduction to curvature of surfaces – video from shape analysis course

Definition of the Gauss map: video.

Differential of the Gauss map

It is a symmetric endomorphism. It also gives the Second fundamental form

• Principal curvatures, $k_1$, $k_2$ given by its eigenvalues; the eigendirections give the Principal directions
• Gaussian curvature, $K$: given by it's Determinant
• Mean curvature, $H$: given by it's Trace (linear operator) (divided by $2$)
• Line of curvature

vid, mean and Gaussian curvature

Smoothness properties of these curavtures

Examples:

• the Plane (surface), vid
• the Sphere, vid. It has constant curvature equal to $1/r^2$
• the Cylinder, video. It has 0 Gauss curvature.

The mean curvature change sign when changing the sign of the normal vector field, while the Gauss curvature doesn't. So in this sense, the sign of the Gauss curvature is geometrically meaningful, while the sign of the mean curvature isn't.

Matrix representation of fundamental forms and differential of Gauss map

Using the standard basis of the Tangent space, given by the Coordinate chart $X$: $X_u$, and $X_v$, where $u$ and $v$ are the coordinates in the domain of $X$

–>video. $I_p$ –> $M$, $II_p$ –> $\Sigma$, and $-dN_p$ –> $A$, where the r.h.s. are the matrices. Note that $MA = \Sigma$, because $II_p$ is works by operating on a vector with $-dN_p$, which in the canonical basis means multiplying with $A$, and then taking the inner product of the resulting vector with another vector, which in the canonical basis means using multiplying the vectors together with $M$ in the middle. The total operation in the canonical basis is $MA$ and is what we call $\Sigma$.

The elements of these matrices are given standard names: $E$, $F$, $G$ for the elements of $M$, and $e$, $f$, $g$ for the elements of $\Sigma$. Then, because $A=M^{-1} \Sigma$, we can obtain the eigenvalues of $A$, which are principal curvatures, in terms of the these standard functions, obtaining for the Gaussian curvature, and for the mean curvature

Examples:

• Elliptic paraboloid
• Helicoid

Geometric interpretation of Gaussian curvature

For points with positive curvature, the surface, locally, lies on one side of the Tangent plane

For points with negative curvature, the surface, locally, lies on both sides of the Tangent plane

To study this analytically, we need to define the Hessian, and the Height function for the tangent plane around any point

Formal formulation

Note that for Elliptic points, the level curves are locally Ellipses, and for Hyperbolic points they are locally Hyperbolas

There is no such theorem for K(p)=0

If S is compact, then there are elliptic points

—> Gauss vs mean curvature, and relating them to the two fundamental forms! Indeed the 1st fundamental form is the Metric tensor

General curvature of Manifolds

See Iham's book, or some books on General relativity, as curvature appears in Einstein's equations

Riemann curvature