Learn geometry and topology: metric spaces, manifolds, continuity and quizzes
154
Sub Topics
204
MCQs
90
MCOs
144
True/False
90
Fill Blanks
32
Rearrange
75
Matching
32
Comprehensions
78
Flashcard Decks
Curriculum
What You'll Learn
01 Foundations of Geometry 3 topics
1 Euclidean Geometry
- Axioms and Postulates
- Basic Constructions
- Congruence and Similarity
- Coordinate Systems
2 Non-Euclidean Geometries
- Spherical Geometry
- Hyperbolic Geometry
- Comparison of Geometries
3 Analytic Geometry
- Cartesian Coordinates
- Vectors and Vector Spaces
- Equations of Lines and Planes
- Conics and Quadrics
02 Differential Geometry 4 topics
1 Curves in Space
- Parametrization
- Arc Length and Reparametrization
- Frenet-Serret Apparatus
- Curvature and Torsion
2 Surfaces in 3D Space
- Parametrized Surfaces
- Tangent Plane and Normal Vector
- First Fundamental Form
- Area and Surface Integrals
3 Curvature of Surfaces
- Second Fundamental Form
- Principal, Gaussian, and Mean Curvatures
- Theorema Egregium
- Geodesics
4 Intrinsic Geometry
- Gauss-Bonnet Theorem
- Parallel Transport
- Covariant Derivative
- Christoffel Symbols
03 Manifolds 4 topics
1 Topological Manifolds
- Definition and Examples
- Charts and Atlases
- Smooth Structures
- Orientability
2 Tangent and Cotangent Spaces
- Tangent Vectors and Bundles
- Differential Forms
- Vector Fields
- Lie Derivatives
3 Riemannian Geometry
- Riemannian Metrics
- Connections and Parallel Transport
- Geodesics and Completeness
- Curvature Tensor
4 Submanifolds
- Embeddings and Immersions
- Normal Bundle
- Second Fundamental Form
- Mean Curvature Vector Field
04 Point-Set Topology 4 topics
1 Topological Spaces
- Open and Closed Sets
- Basis and Subbasis
- Subspaces and Product Spaces
- Quotient Spaces
2 Continuity and Homeomorphisms
- Continuous Functions
- Homeomorphisms
- Topological Invariants
- Embeddings
3 Separation Axioms
- T0, T1, and Hausdorff Spaces
- Regular and Normal Spaces
- Completely Regular Spaces
- Metrization Theorems
4 Compactness and Connectedness
- Compact Spaces
- Connected and Path-Connected Spaces
- Locally Compact Spaces
- Paracompactness
05 Algebraic Topology 4 topics
1 Fundamental Group
- Homotopy
- Fundamental Group Definition
- Covering Spaces
- Van Kampen's Theorem
2 Homology Theory
- Simplicial Complexes
- Chain Complexes
- Simplicial Homology
- Singular Homology
3 Cohomology
- Cochains and Coboundaries
- Universal Coefficient Theorem
- Cup Product
- Poincaré Duality
4 Homotopy Theory
- Higher Homotopy Groups
- Fibrations and Fiber Bundles
- Obstruction Theory
- Spectral Sequences
06 Differential Topology 4 topics
1 Smooth Manifolds
- Differentiable Structure
- Partitions of Unity
- Tangent and Cotangent Bundles
- Vector Bundles
2 Transversality
- Transverse Maps
- Sard's Theorem
- Transversality Theorems
- Applications
3 Morse Theory
- Critical Points and Morse Functions
- Morse Lemma
- Handle Decomposition
- Morse Inequalities
4 Characteristic Classes
- Stiefel-Whitney Classes
- Euler Class
- Pontryagin Classes
- Chern Classes
07 Low-Dimensional Topology 4 topics
1 Surfaces
- Classification of Surfaces
- Euler Characteristic
- Mapping Class Groups
- Teichmüller Theory
2 3-Manifolds
- Prime Decomposition
- Geometric Structures
- Thurston's Geometrization
- Knot Theory
3 4-Manifolds
- Intersection Form
- Signature and Euler Characteristic
- Exotic Structures
- Donaldson and Seiberg-Witten Invariants
4 Surgery Theory
- Handle Decompositions
- Surgery Techniques
- Kirby Calculus
- h-Cobordism Theorem
08 Geometric Group Theory 4 topics
1 Groups Acting on Spaces
- Proper and Cocompact Actions
- Orbit Spaces and Quotients
- Discrete Groups of Isometries
- Lattices in Lie Groups
2 Hyperbolic Groups
- Cayley Graphs
- Quasi-Isometries
- Hyperbolicity
- Boundary at Infinity
3 Geometric Invariants
- Growth Functions
- Isoperimetric Inequalities
- Dehn Functions
- Dimension Theory
4 Rigidity Phenomena
- Mostow Rigidity
- Superrigidity
- Quasi-Isometric Rigidity
- Bounded Cohomology
09 Symplectic and Contact Geometry 4 topics
1 Symplectic Manifolds
- Symplectic Forms
- Darboux's Theorem
- Moment Maps
2 Hamiltonian Mechanics
- Hamilton's Equations
- Poisson Brackets
- Integrable Systems
- Action-Angle Coordinates
3 Contact Manifolds
- Contact Structures
- Darboux's Theorem for Contact Forms
- Legendrian Submanifolds
- Reeb Vector Fields
4 Symplectic Topology
- Gromov's Non-Squeezing Theorem
- Floer Homology
- Pseudoholomorphic Curves
- Symplectic Capacities
10 Advanced Topics and Applications 4 topics
1 Geometric Analysis
- Elliptic Operators
- Heat Equation
- Hodge Theory
- Index Theorems
2 Geometric Flows
- Mean Curvature Flow
- Ricci Flow
- Yamabe Flow
- Kähler-Ricci Flow
3 Quantum Topology
- Jones Polynomial
- Topological Quantum Field Theories
- Reshetikhin-Turaev Invariants
- Khovanov Homology
4 Applied Topology
- Computational Topology
- Persistent Homology
- Topological Data Analysis
- Applications in Sciences
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Geometry/Topology
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