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Geometry

Plane Figures

Solid Figures

Generalizations

Differential Geometry

Local Curve Theory

Global Curve Theory

Local Surface Theory

In surface theory, the domain of a simple surface x:UR3\mathbf{x} : U \to \R^{3} is denoted by coordinates (u1,u2)(u_{1}, u_{2}) or (u,v)(u,v). Einstein notation is actively used in cases of (u1,u2)(u_{1}, u_{2}). When avoiding messy notation with unnecessary subscripts, (u,v)(u, v) is used.

First and Second Fundamental Forms

Geodesics and Parallelism

Weingarten MapShape Operator

Curvature

Fundamental The

orem of Surfaces

Surfaces of Constant Curvature

Global Surface Theory

Simple Curvature

Orientability

Gauss-Bonnet Theorem

Jacobi’s Theorem

  • Jacobi’s Theorem

Index of Vector Fields

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