Euclidean and non-Euclidean geometry: axioms of point/line/plane, distance and angle, the parallel postulate, and their consequences. Bridges into coordinate geometry (Descartes). Topology-space and differential-geometry are re-parented to…
geometry
Euclidean axioms
The five postulates of Euclid's Elements (Book I): point/line/extension, any segment extends to a line, any point+radius define a circle,…
Parallel postulate
Through any point not on a given line there is exactly one line parallel to it (Playfair form). Its independence — proven in the 19th…
Point
Undefined primitive of Euclidean geometry: a location with no extent.
Line
Undefined primitive of Euclidean geometry: a breadthless length extending without end in either direction.
Triangle inequality
For any triangle, the sum of any two sides is strictly greater than the third: a + b > c. Generalises to any metric space.
Sum-of-angles (Euclidean triangle)
The interior angles of any Euclidean triangle sum to π radians (180°). Equivalent to the parallel postulate.
Pythagorean theorem (geometry view)
In any Euclidean right triangle, a² + b² = c², where c is the hypotenuse. Equivalent formulation of the parallel postulate; fails in…
Circle and π ratio
A circle is the locus of points equidistant from a centre. The ratio of its circumference to its diameter is the transcendental constant π.
Coordinate geometry (Cartesian)
Descartes' bridge between geometry and algebra: every point in the plane is a pair (x, y) ∈ ℝ², every line is a linear equation, every…
Non-Euclidean geometries
Hyperbolic (Lobachevsky, Bolyai) and spherical geometries: drop or replace the parallel postulate and angle sums become < π (hyperbolic) or…
K (Kepler–Bouwkamp constant)
The limit of the product of the ratios inradius/circumradius over successive regular k-gons inscribed in the previous circle, for k = 3, 4,…
P (universal parabolic constant)
Ratio of parabolic arc length to the parabola's latus rectum. Universal — same for every parabola.
Smooth manifold
A Hausdorff, second-countable topological space locally homeomorphic to ℝ^n with C^∞-compatible charts. Setting for differential geometry,…
Tangent space T_p M
At each point p of a smooth n-manifold, an n-dimensional vector space of derivations C^∞(M)→ℝ at p (equivalently, equivalence classes of…
Vector field
A smooth section of the tangent bundle TM → M. Integrates to flows; infinitesimal generators of diffeomorphism groups.
Differential form
A smooth section of Λ^k T*M — a totally-antisymmetric k-linear functional on tangent vectors at each point. Objects of integration on…
Riemannian metric
A smooth, symmetric, positive-definite (0,2)-tensor g on a manifold M. Determines lengths, angles, geodesics, curvature, and volume.
Geodesic
A locally length-minimising (or critical-length) curve γ on a Riemannian manifold, satisfying the geodesic equation ∇_{γ'} γ' = 0. …
Riemann curvature tensor
The (1,3)-tensor R(X,Y)Z = ∇_X∇_Y Z − ∇_Y∇_X Z − ∇_{[X,Y]} Z measuring failure of commutativity of covariant derivatives. Contracts to…
Gauss–Bonnet theorem
For a compact oriented Riemannian 2-manifold M without boundary, ∫_M K dA = 2π χ(M). Connects local curvature to global topology;…
Symplectic manifold
A 2n-manifold M equipped with a closed, non-degenerate 2-form ω. Underlies Hamiltonian mechanics; Darboux: locally all symplectic…
Kähler manifold
A complex manifold with a Hermitian metric whose associated 2-form is closed (equivalently the complex structure is parallel). Combines…
Levi-Civita connection
Unique torsion-free metric-compatible connection ∇. Christoffel symbols Γ^k_{ij} = ½g^{kl}(∂_i g_{jl} + ∂_j g_{il} − ∂_l g_{ij}).
Riemann curvature tensor
R(X,Y)Z = ∇_X ∇_Y Z − ∇_Y ∇_X Z − ∇_{[X,Y]}Z. Ricci Ric = tr R, scalar S = tr Ric. Sectional curvature K(X∧Y).
Sectional / Ricci / scalar curvature
Scalar invariants from R. Positive sectional: Myers' theorem bounds diameter; Ricci lower bound: volume comparison (Bishop-Gromov).
Gauss-Bonnet (general)
On closed oriented even-dim Riemannian M: ∫_M Pf(Ω) = χ(M) · (2π)^n / n!. Chern's intrinsic proof 1944. Atiyah-Singer generalization.
Einstein manifold
Ric = λg for constant λ. Models Einstein vacuum with cosmological constant. Examples: spheres, Kähler-Einstein, Calabi-Yau (Ric=0).
Calabi-Yau manifold
Compact Kähler with c_1 = 0. Yau 1977 theorem: admits unique Ricci-flat metric. Compactification manifold in string theory.
Symplectic manifold
Pair (M,ω) with closed non-degenerate ω ∈ Ω²(M). Darboux: locally (ℝ^{2n}, dp∧dq). Hamiltonian flows; moment maps; Gromov's non-squeezing.
Contact manifold
Odd-dim M with 1-form α, α ∧ (dα)^n ≠ 0. Reeb vector field; contactomorphisms rigid (Gromov), Eliashberg-Hofer theory.
Principal G-bundle
Fibre bundle P → M with free G-action, fibre ≅ G. Connection 1-form ω: TP → g. Yields gauge theory framework.
Ehresmann connection & curvature form
Horizontal distribution H ⊂ TP, G-equivariant. Curvature Ω = dω + ½[ω,ω] ∈ Ω²(P,g). Flat: Ω = 0; parallel transport integrable.
Chern classes
c(E) = 1 + c_1(E) + c_2(E) + … topological invariants of complex vector bundle. Chern-Weil: c_k = [tr(F^k)] from curvature F.
Hopf fibration
S¹ → S³ → S² (complex); quaternionic S³ → S⁷ → S⁴; octonionic S⁷ → S¹⁵ → S⁸. Generators of π_3(S²), etc.
Hyperbolic geometry (ℍⁿ)
Constant sectional curvature −1. Models: upper half-plane, Poincaré disk, hyperboloid. Isometry group PSL(2,ℝ); parallel postulate fails.
Spherical geometry Sⁿ
Constant +1 curvature. Isometries O(n+1). Spherical triangle excess = area. Conjugate points; diameter = π; total geodesic closed.
Ricci flow (Hamilton-Perelman)
∂_t g = −2 Ric(g). Hamilton 1982; Perelman 2003 solved Poincaré conjecture via entropy monotonicity + surgery.
Minimal surface
Mean curvature H = 0: critical point of area functional. Plateau's problem (soap films). Bernstein's theorem ℝⁿ (n ≤ 7); Simons' inequality.
Geometric measure theory
Currents, varifolds (Federer, Almgren). Rectifiable sets; regularity of area-minimising currents. Solves Plateau in all dimensions.
Law of cosines
For any triangle with sides a, b, c and the angle C opposite to c, the law of cosines states c² = a² + b² − 2ab cos(C). Recovers the…
Law of sines (circumradius form)
For any triangle with sides a, b, c, opposite angles A, B, C, and circumradius R, a/sin A = b/sin B = c/sin C = 2R. The common ratio…
Heron's formula
Area of a triangle from its sides: K = √(s(s−a)(s−b)(s−c)), where s is the semiperimeter. Derived from the law of cosines and the identity…
Euler's polyhedron formula
For any convex polyhedron, V − E + F = 2. More generally χ = V − E + F is the Euler characteristic of the underlying 2-sphere, and χ = 2 −…
Classification of Platonic solids
There are exactly 5 regular convex polyhedra (Euclid's Elements XIII): the tetrahedron {3,3}, octahedron {3,4}, icosahedron {3,5}, cube…
Cross-ratio invariance
The cross-ratio (z₁, z₂; z₃, z₄) of four distinct points on a projective line P¹ is the fundamental projective invariant: it is preserved…
Menelaus' theorem
For a triangle ABC and a transversal line meeting BC, CA, AB at X, Y, Z (with points extended if necessary), the product of signed ratios…
Ceva's theorem
For triangle ABC with cevians AD, BE, CF (D ∈ BC, E ∈ CA, F ∈ AB), the three cevians are concurrent iff (BD/DC)(CE/EA)(AF/FB) = +1 (signed…