Geometry
Plane Figures
- Straight Lines
- Triangles
- Circle
- Conic Sections, Conical Curves
Solid Figures
Generalizations
- Definition of Basis Orientation in Vector Spaces
- General Definitions of Angles and Perpendicularity
- General Definitions of Lines, Planes, and Spheres
- General Definition of a Spiral
- General Definition of a Parallelepiped
- Moduli Space
- Definition of Convex Hull
- Simplex $\Delta^{n}$
Differential Geometry
Local Curve Theory
- Tangent and Tangent Vector Field
- Definition of Arc
- Frenet-Serret Apparatus: Curvature, Tangent, Normal, Binormal, Torsion
- Frenet-Serret Formulas
- 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 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 Transformation
- Tangent Plane and Normal Vector
- Tangent Vector
- Parameterized Curves on a Simple Surface
- Intrinsic Mapping
- $C^{k}$ Surfaces
- Surface of Revolution
First and Second Fundamental Forms
- First Fundamental Form
- Normal Curvature and Geodesic Curvature $\kappa_{n}$, $\kappa_{g}$
- Second Fundamental Form
- Christoffel Symbols $\Gamma_{ij}^{k}$
- Gauss’s Theorema Egregium
- ‘Intrinsic/Essential’ Definition
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 MapShape 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 The
orem of Surfaces
Surfaces of Constant Curvature
- Classification of Surfaces of Revolution According to Gauss Curvature
- $K \gt a^{2}:$ Surfaces of Positive Curvature
- $K = 0:$ Surfaces of Zero Curvature
- $K \lt -a^{2}:$ Surfaces of Negative Curvature
Global Surface Theory
Simple Curvature
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
- Differentiation of Functions Defined on Manifolds $d \phi_{p}$
- Differential Isomorphism
- Immersion and 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 $R$
- Sectional Curvature $K(\sigma)$
- 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)
All posts
- Derivation of the Formula to Calculate the Distance Between Two Parallel Lines
- Area of a Triangle Enclosed by a Straight Line and the x and y Axes
- Equation of the Tangent to a Parabola
- Prove that the Product of the Slopes of Two Perpendicular Lines is Always -1
- Finding Internal and External Division Points on a Line
- Calculating the Area of an Ellipse Using Integration
- Pythagorean Theorem Proof
- Equation of the Tangent to a Circle with Slope m
- Finding the Equation of the Tangent Line at a Point on a Circle
- Properties of a Line Passing Through the Focus of a Parabola
- Proof that the Length of an Arc and the Length of a Chord are Approximately Equal When the Central Angle is Small
- Definition of a General Parallelepiped
- Derivation of the Equation of an Ellipse
- Ellipse
- Perimeter of an Ellipse
- Area of a Surface in Differential Geometry
- Classification of Surfaces of Revolution According to Gaussian Curvature
- Properties of Rotational Surfaces
- Translation of Vector Field Translation:
- Solid Angle of a Sphere
- Definition of General Angles and Perpendicularity
- Basis Orientation Defined in a Vector Space
- General Definitions of Lines, Planes, and Spheres
- Definition of a Curve
- Reparameterization
- Tangent Lines and Tangent Vector Fields
- The Definition of Strings
- Frenet-Serret Formulas: Curvature, Tangent, Normal, Binormal, Torsion
- Frenet-Serret Formulas
- Conditions for a Curve to Lie in a Plane in Three-Dimensional Euclidean Space
- Definition of a General Spiral
- Lanczos Theorem Proof
- Tangent Plane and Normal Plane
- Formulas for Curves on a Sphere
- Differentiable Manifolds
- Reparameterization and the Tools of Frenet-Serret
- Proof of the Fundamental Theorem of Curves
- Coordinates of a Three-Dimensional Unit Sphere
- Tangents, Normals, and Curvature of Plane Curves
- Definition of a Closed Curve
- Definition of a Simple Curve
- Moduli Spaces
- Winding Number of a Closed Curve
- Vector Space
- Rotation Number of Plane Curves
- Differentiable Functions from a Differentiable Manifold to a Differentiable Manifold
- Proof of the Rotation Number Theorem
- Tangent Vector on Differentiable Manifold
- Derivation of the Area Formula for a Region Enclosed by a Simple Closed Plane Curve
- Proof of the Rearrangement Inequality
- Differentiation of Functions Defined on Differentiable Manifolds
- Simple Surfaces, Coordinate Mapping
- Coordinate Transformation in Curved Surface Theory
- Differentiable Homomorphism
- Intersection of a Plane and a Normal Vector
- Tangent Vectors on Simple Surfaces
- Eigen Decomposition
- Definition of Surfaces in Differential Geometry
- First Basic Forms, Riemannian Metrics
- Specific Examples of Calculations Using the Riemann Metric
- Parametric Curves on a Simple Surface
- Gauss Curvature and Geodesic Curvature
- Second Fundamental Form in Differential Geometry
- Christoffel Symbols in Differential Geometry
- Gauss's Theorem in Differential Geometry
- Definition of Intrinsic in Differntial Geometry
- The Christoffel Symbols are Intrinsic
- The Tangent Space on an n-Dimensional Differentiable Manifold is an n-Dimensional Vector Space
- Geodesic Curvature is Intrinsic
- Definition of a Straight Line (Geodesic) in Differential Geometry
- Rotational Surfaces in Differential Geometry
- Emersion and Embedding on a Differential Manifold
- Geodesic on a Surface of Revolution
- Definition of Parallel Vector Field along a Curve on Surface
- Uniqueness Theorem of Geodesics
- If It's the Shortest Curve, It's a Geodesic
- Directional Derivatives in Differential Geometry
- Geodesic Coordinate Transformation
- Properties of the Second Normal Form
- Definition of Normal Sections and Menelaus's Theorem
- Bingarten Map
- Bingarten Equation
- The Relationship between the Second Normal Form and the Vingarten Map
- The Relationship between the Fundamental Form and Coordinate Transformation
- Curvature of a Principal Curve
- Euler's Theorem in Differential Geometry
- Properties of Vector Fields Parallel to a Curve
- Gaussian Curvature and Mean Curvature
- Definition and Relationship between the Gaussian Map and Gaussian Curvature
- Necessary and Sufficient Conditions for a Vector Field to be Parallel Along a Curve
- Riemann Curvature Tensor, Gauss Equation, and Codazzi-Mainardi Equation in Differential Geometry
- Gauss's Great Theorem
- Differentiable Functions Between Two Surfaces
- Differential Geometry: Isometric Mappings
- Differentiable Geometry: Local Isometries
- Fundamental Theorem of Curved Surfaces
- Positive Gaussian Curvature Surfaces of Revolution
- Two Rotational Surfaces with Positive Curvature are Locally Isometric
- The Rotational Surface with Zero Curvature
- Differentiable Manifolds in Compact Surfaces
- Gaussian Curvature
- Differentiable Surfaces and Boundaries of Regions in Differential Geometry
- Homotopy to Null in Differential Geometry
- Simple Connected Region
- Gauss-Bonnet Theorem
- Euler Characteristics in Geometry
- Cotangent Space and First-Order Differential Forms
- Second-Order Differential Form
- kth Order Differential Forms
- Definition of Convex Hull
- Operations on Differential Forms: Sum and Wedge Product
- Pull Back in Differential Geometry
- Differential Forms of Type k
- Emulsions are locally embedded.
- Tangent Bundles on Differentiable Manifolds
- Vector Field on Differentiable Manifold
- Lie Brackets of Vector Fields
- Riemann Metric and Riemann Manifolds
- Isometries and Local Isometries on Riemann Manifolds
- Vector Fields Along Curves on Differential Manifolds
- Affine Connection
- Covariant Derivative of Vector Fields
- Parallel Vector Fields on Differential Manifolds
- Coexistence Compatible Connection
- Symmetry of Connection
- Levi-Civita Connection, Riemannian Connection, Coefficients of Connection, Christoffel Symbols
- Geodesics on a Differentiable Manifold
- Geodesic Flow
- Homogeneity of Geodesics
- Exponential Mapping
- Differentiable Curves and Minimization
- Parameterized Surfaces
- Gauss's Lemma in Riemannian Geometry
- Minimizing Geodesics
- Exponential Mapping and Normal Neighborhood
- Poincaré Metric
- Differential Geometry of Curved Manifolds
- Sectional Curvature of Differential Manifolds
- Bianchi Identity
- Symmetry of the Riemann Curvature Tensor
- Coordinate Representation of the Riemann Curvature Tensor
- If the Sectional Curvature is the Same, the Riemannian Curvature is also the Same.
- Ricci Curvature of Differentiable Manifolds
- Scalar Curvature of Differential Manifolds
- The Relationship between Covariant Derivative and Riemann Curvature Tensor
- Tensors Defined on Differentiable Manifolds
- Differentiable Vector Fields on a Differentiable Manifold
- Sets of Differentiable Real-Valued Functions on a Differentiable Manifold
- Definition of Simplex
- Gaussian Curvature with Negative Values on Rotational Surfaces
- The Coordinate Patch Mapping of a Torus in Three-Dimensional Space
- Total Variation in Differential Geometry
- Parabola
- Hyperbola
- Definition of Circle
- Quadratic Curve
- Geodesic Coordinate Patch Mapping and Christoffel Symbols
- Geodesic Coordinate Mapping and Gaussian Curvature
- Coordinates of a Möbius Strip in Three-Dimensional Space