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Geometry

Axiomatic Geometry

This will not be limited strictly to what can be called axiomatic geometry, but will cover content ranging from middle school curriculum to about the level of a second-year undergraduate.

Plane Figures

Solid Figures

Differential Geometry

Local Curve Theory

Global Curve Theory

Local Surface Theory

In surface theory, the domain of a simple surface $\mathbf{x} : U \to \R^{3}$ is denoted by coordinates $(u_{1}, u_{2})$ or $(u,v)$. Einstein notation $a_{i}b_{i} = \sum_{i}a_{i}b_{i}$ is actively used in cases of $(u_{1}, u_{2})$. When avoiding messy notation with unnecessary subscripts, $(u, v)$ is used.

First and Second Fundamental Forms

Geodesics and Parallelism

Weingarten MapShape Operator

Curvature

Fundamental Theorem of Surfaces

Surfaces of Constant Curvature

Global Surface Theory

Simple Curvature

Orientability

Gauss-Bonnet Theorem

  • Gauss-Bonnet Formula
  • Global Gauss-Bonnet Formula
  • Euler Characteristic $\chi = V - E + F$
  • Regular Region, Simple Region, Triangulation
  • Definition of Genus and its Relationship with Euler Characteristic $\chi = 2(1-g)$

Jacobi’s Theorem

  • Jacobi’s Theorem

Index of Vector Fields

  • Index of Zeroes of a Vector Field and Index $I(V) = \sum i_{p}(V)$
  • Poincaré-Brouwer Theorem $I(M) = \chi(M)$

Differential Forms

Differential Manifolds and Riemannian Geometry

Vector Fields

Riemannian Metric, Connection

Geodesics

  • Geodesic
  • Flow
  • Homogeneity
  • Exponential Map
  • Differentiable Curves and Minimization
  • Parameterized Surfaces
  • Gauss’s Lemma
  • Minimizing Properties of Geodesics
  • Exponential Map and Normal Neighborhood
  • Poincaré Metric

Curvature

References

  • Richard S. Millman and George D. Parker, Elements of Differential Geometry (1977)
  • Manfredo P. Do Carmo, Riemannian Geometry (Eng Edition, 1992)

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