Differential Geometry notes

By Xah Lee. Date: . Last updated: .

Differential geometry

Differential geometry

Differential geometry of curves

Differential geometry of surfaces

tangent space

for each point on a manifold, you have a tangent space.

once tangent space is defined, you can define a vector field. That is, pick a vector in tangent space at point p, and for any point p. So now, each point has a vector, thus vector field.

vector field

tangent bundle

Atlas (topology)


jacobian matrix, derivative of multi-valued function

if f maps R2 to R3, we take partial derivative of each argument.

so, we got a new function f' that is also R2 to R3.

let's say our f is

 {f u v} =
 (u^2 + v^2)
 (u + v)
 (u * v)

local vs global

The property: A function f is bounded has a local and a global version:

The property: “A function f is continuous” does not have a local and a global version because: A function is called continuous on a space X if, for each point p of X, it is continuous at p.

The property: "A function f is uniformly continuous" has a local and a global version on metric spaces.

Gaussian curvature

Gaussian curvature

Gauss map

In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N(p) is a unit vector orthogonal to X at p, namely the normal vector to X at p.

Riemann surface

particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and real projective plane do not.

Geometrical facts about Riemann surfaces are as “nice” as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The Riemann–Roch theorem is a prime example of this influence.

let us try to settle this “family” business:

Our simpler minimal surfaces are defined as images of the following kind: First step consider

F : C union {point at infinity} → C^3 union {some infinite points = the images of the punctures}

We need two conditions on F.

then: The image of real(F) is a minimal surface in R^3.

Almost of course it is then also true that F_a := real( exp(i*a) F) : C ⟶ R^3 has also minimal surfaces as images.

This family of minimal surfaces (family parameter a) is called the associate family of the first minimal surface, which is given as the image of real(F) .

Gauss–Bonnet theorem

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