Differential Geometry notes
- Curves in R
- n 2
- 1.1. Parametrized curves 2
- 1.2. arclength parameter 2
- 1.3. Curvature of a plane curve 4
- 1.4. Some elementary facts about inner product 5
- 1.5. Moving frames along a plane curve 7
- 1.6. Orthogonal matrices and rigid motions 8
- 1.7. Fundamental Theorem of plane curves 10
- 1.8. Parallel curves 12
- 1.9. Space Curves and Frenet frame 12
- 1.10. The Initial Value Problem for an ODE 14
- 1.11. The Local Existence and Uniqueness Theorem of ODE 15
- 1.12. Fundamental Theorem of space curves 16
- 2. Fundamental forms of parametrized surfaces 18
- 2.1. Parametrized surfaces in R
- 3 18
- 2.2. Tangent and normal vectors 19
- 2.3. Quadratic forms 20
- 2.4. Linear operators 21
- 2.5. The first fundamental Form 23
- 2.6. The shape operator and the second fundamental form 26
- 2.7. Eigenvalues and eigenvectors 28
- 2.8. Principal, Gaussian, and mean curvatures 30
- 3. Fundamental Theorem of Surfaces in R
- 3 32
- 3.1. Frobenius Theorem 32
- 3.2. Line of curvature coordinates 38
- 3.3. The Gauss-Codazzi equation in line of curvature coordinates 40
- 3.4. Fundamental Theorem of surfaces in line of curvature
- coordinates 43
- 3.5. Gauss Theorem in line of curvature cooridnates 44
- 3.6. Gauss-Codazzi equation in local coordinates 45
- 3.7. The Gauss Theorem 49
- 3.8. Gauss-Codazzi equation in orthogonal coordinates 51
Differential geometry
Differential geometry of curves
Differential geometry of surfaces
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.
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} =
{
list
(u^2 + v^2)
(u + v)
(u * v)
}
local vs global
- A property is called “locally true” if every point of your space has a neighborhood on which the property is true.
- A property is “globally true” if it is true at all points of your space.
The property: A function f is bounded has a local and a global version:
- f is locally bounded if for every point of your space there is a neighborhood on which you have a bound for f.
- f is globally bounded if you have a bound for the values of f at all points of your space.
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.
- local: Every point p in X has a neigborhood on which f is uniformly continuous.
- global: f is uniformly continuous on X.
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.
- F is complex differentiable
- real(F) : C → R^3 is conformal onto its 2-dim image.
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) .