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.

• 🔒(25/10/14)Euclidean Axiom System
• 🔒(25/10/16)Hilbert Axiom System
• 🔒(25/10/18)Birkhoff Axiom System

Plane Figures

• 🔒(25/10/20)Straight Line
  • Formula for the distance between two parallel lines $d=\frac { |2k| }{ \sqrt { m^{ 2 }+1 } }$
  • The product of the slopes of two perpendicular lines is always -1
  • Internal and external division points on a perpendicular line
  • 🔒(25/10/26)Deriving the Formula for the Coordinates of the Foot of the Perpendicular on a Plane
• 🔒(25/10/22)Triangle
  • Pythagorean Theorem
  • Area of a triangle enclosed by a straight line and the x-axis and y-axis
• Quadrilateral
  • Square
  • Rectangle
  • Rhombus
  • Parallelogram
  • Trapezoid
• Circle
  • Radian
  • Equation of a tangent line with slope $m$
  • Equation of a tangent line at a point on a circle
  • When the central angle is small, the arc length and chord length are approximately equal
• Quadratic Curves, Conic Sections
  • Parabola
    • Derivation of the equation of a tangent line to a parabola
    • Property of a line passing through the focus of a parabola
  • Ellipse
    • Derivation of the equation of an ellipse
    • Circumference of an ellipse
    • Finding the area of an ellipse using integration
  • Hyperbola

Solid Figures

• Solid angle of a sphere
• Parallelepiped
  • Volume formula for a parallelepiped
• Orientation of a basis defined in a vector space
• General definition of angle and perpendicularity
• General definition of straight line, plane, and sphere
• General definition of a helix
• General definition of a parallelepiped
• Moduli space
  • Projective space
• Definition of convex hull
• Simplex $\Delta^{n}$
• Point of tangency and point of intersection

Differential Geometry

• Definition of a Curve
  • Reparameterization
• Definition of a Closed Curve
  • Rotation Number of a Closed Curve
• Definition of a Simple Curve

Local Curve Theory

• Tangent and Tangent Vector Field
• Definition of Arc
• Frenet-Serret Apparatus: Curvature, Tangent, Normal, Binormal, Torsion
  • Tangent Plane and Normal Plane
• Frenet-Serret Formulas
  • Condition for a Curve to Lie in a Plane in 3D Euclidean Space
  • Lancret’s Theorem
  • Formulas for Curves on a Sphere
• Reparameterization and Frenet-Serret Tools
• Fundamental Theorem of Curves

Global Curve Theory

• Tangent, Normal, and Curvature of Plane Curves
• Rotation Number of Plane Curves
• Rotation Number Theorem
• Area Formula for Regions Enclosed by Simple Closed Planar Curves
• Isoperimetric Inequality

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.

• Simple Surface, Coordinate Patch Mapping
  • Coordinate Patch Mapping of a 3D Unit Sphere
  • Coordinate Patch Mapping of a Torus in 3D Space
  • Coordinate Patch Mapping of a Möbius Strip in 3D Space
• Coordinate Transformation
• Tangent Plane and Normal Vector
• Tangent Vector
• Parameterized Curves on a Simple Surface
• Intrinsic Mapping
• $C^{k}$ Surfaces
• Surface of Revolution
  • Properties

First and Second Fundamental Forms

• First Fundamental Form
  • Concrete Examples
  • Relationship between the First Fundamental Form and Coordinate Transformation
• Normal Curvature and Geodesic Curvature $\kappa_{n}$, $\kappa_{g}$
• Second Fundamental Form
  • Properties
• Christoffel Symbols $\Gamma_{ij}^{k}$
• Gauss’s Theorema Egregium
• ‘Intrinsic/Essential’ Definition
  • Christoffel Symbols are Intrinsic
  • Geodesic Curvature is Intrinsic

Geodesics and Parallelism

• Geodesic
• Geodesics on Surfaces of Revolution
• Uniqueness
• A Curve is a Geodesic if it is the Shortest Path
• Parallel Vector Fields Along a Curve
• Necessary and Sufficient Condition for a Parallel Vector Field Along a Curve
• Parallel Transport of a Vector Field
• Properties

Weingarten Map^{Shape Operator}

• Directional Derivative $\mathbf{X}f$
• Definition of Normal Section and Meusnier’s Theorem
• Weingarten Map $L$
• Weingarten Equation
• Relationship between the Second Fundamental Form and the Weingarten Map

Curvature

• Principal Curvatures $\kappa_{1}, \kappa_{2}$
• Euler’s Theorem
• Gauss Curvature and Mean Curvature $K = \kappa_{1}\kappa_{2}, H = \dfrac{\kappa_{1} + \kappa_{2}}{2}$
• Surface Area
• Definition of Gauss Map and its Relationship with Gauss Curvature
• Riemann Curvature Tensor $R_{ijk}^{l}$, Gauss’s Theorema Egregium, Codazzi-Mainardi Equations
• Gauss’s Remarkable Theorem

Fundamental Theorem of Surfaces

• Differentiable Function Between Two Surfaces
• Isometry
• Local Isometry
• Fundamental Theorem of Surfaces

Surfaces of Constant Curvature

• Classification of Surfaces of Revolution According to Gauss Curvature
• $K \gt a^{2}:$ Surfaces of Positive Curvature
  • Two Surfaces of Positive Curvature are Locally Isometric
• $K = 0:$ Surfaces of Zero Curvature
• $K \lt -a^{2}:$ Surfaces of Negative Curvature

Global Surface Theory

• Geodesic Coordinate Patch Mapping
  • Christoffel Symbols
  • Gauss Curvature

Simple Curvature

• Compact Surfaces
• Conditions for a Compact Surface to be a Sphere

Orientability

• Orientable Surfaces
• Region of a Surface and Boundary of a Region
• Total Angular Variation
• Null Homotopic
• Simply Connected Region

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

• Cotangent Space $T_{p}^{\ast}M$, $1$st Order Differential Forms $\omega : M \to T_{p}^{\ast}M$
• $2$nd Order Differential Forms $\omega : M \to \Lambda^{2} (T_{p}^{\ast}M)$
• $k$th Order Differential Forms $\omega : M \to \Lambda^{k} (T_{p}^{\ast}M)$
• Operations on Differential Forms: Sum and Wedge Product $\wedge$
• Pullback
• Exterior Derivative of a $k$th Order Differential Form

Differential Manifolds and Riemannian Geometry

• Differentiable Manifold $M$
• Differentiable Functions on Manifolds $f : M_{1} \to M_{2}$
• Set of Differentiable Functions on a Manifold $\mathcal{D}(M)$
• Tangent Vector and Tangent Space $T_{p}M$ on a Differential Manifold
  • Tangent Space on an $n$-dimensional Differential Manifold is an $n$-dimensional Vector Space
• Differentiation of Functions Defined on Manifolds $d \phi_{p}$
• Differential Isomorphism
• Immersion and Embedding
  • Immersion is Locally an Embedding

Vector Fields

• Tangent Bundle $TM = \bigcup\limits_{p \in M}T_{p}M$
• Vector Field $X : M \to TM$
• Set of Differentiable Vector Fields on a Manifold $\frak{X}(M)$
• Lie Bracket $[X, Y] = XY - YX$
• Local Flow

Riemannian Metric, Connection

• Riemannian Metric and Riemannian Manifold $(M, g)$
• Isometries and Local Isometries
• Vector Fields Along Curves
• Affine Connection $\nabla_{X}Y$
• Covariant Derivative $\dfrac{d V}{d t}$
• Parallel Vector Fields
• Compatible Connection
• Symmetry of Connections
• Levi-Civita 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

• Riemann Curvature Tensor $R$
  • Bianchi Identity
  • Symmetries
  • Coordinate Representation $R_{ijkl}$
• Sectional Curvature $K(\sigma)$
  • Equal Sectional Curvatures Imply Equal Riemann Curvatures
• Ricci Curvature $\Ric$
• Scalar Curvature $K$
• Relationship between Covariant Derivative and Riemann Curvature
• What is a Tensor? $T : \frak{X}(M) \times \cdots \times \frak{X}(M) \to \mathcal{D}(M)$
• Covariant Differentiation of a Tensor $\nabla T$, Covariant Derivative $\nabla_{Z}T$

References

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