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This metric is the hyperbolic plane with curvature scale $a$. Its nonzero connection coefficients are

A metric tensor is a nondegenerate symmetric covariant tensor that defines squared line elements by $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$.

After using Killing constants and four-velocity normalization, radial timelike geodesic motion can often be written as kinetic energy plus an effective potential equal to a constant.

Rotational symmetry conserves the angular-momentum direction, so every nonradial geodesic lies in a plane through the symmetry centre; coordinates may place that plane at $\theta =\pi /2$.

Anti-de Sitter spacetime is a maximally symmetric spacetime with a Lorentzian metric and constant negative sectional curvature. In four dimensions, curvature radius $L$ corresponds to cosmological constant $\Lambda =-3/L^2$. Its global static geometry has a timelike conformal boundary.

A radial light ray approaches that boundary in finite static-coordinate time but infinite physical affine parameter. Consequently, specifying evolution of a field generally requires suitable boundary conditions as well as initial data. One commonly uses the universal cover to remove the closed timelike time circles of the embedded hyperboloid. This geometry is central to AdS/CFT correspondence, where bulk fields encode observables of a boundary conformal field theory.^[ref]

De Sitter spacetime is the maximally symmetric Lorentzian spacetime of constant positive curvature. Its expanding flat slicing has scale factor $a(\tau )=-1/(H\tau )$ in conformal time $\tau <0$.

The Schwarzschild spacetime describes the spherically symmetric vacuum spacetime with zero cosmological constant. Its exterior chart, in units $G=c=1$ and metric signature $(-+++)$, isThe areal radius $r$ makes each symmetry sphere have area $4\pi r^2$; the mass parameter $M$ has length units in this convention. Birkhoff's theorem makes this the local vacuum geometry outside a spherical body, including when its interior changes spherically. A material star need not extend down to the Schwarzschild radius $2M$.

For the black-hole extension, $r=2M$ is an event horizon. The singular metric coefficients of this chart can be removed with Eddington-Finkelstein coordinates or Painlevé–Gullstrand coordinates. The invariant $R_{abcd}{R}^{abcd}=48M^2/r^6$ is finite there and diverges at $r=0$ in the extension. This Ricci flat metric has nonzero tidal forces. The $M=0$ limit recovers Minkowski spacetime in spherical coordinates.^[ref]

Linearized gravity writes $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ with $|h_{\mu \nu }|\ll 1$ and discards terms quadratic in $h$ and its derivatives.

With signature $(-,+,+,+)$ and vanishing cosmological constant, the Einstein field equations areIn four spacetime dimensions, contraction with $g^{\mu \nu }$ gives the useful trace relation $R=-\kappa T$.

In the usual commutator convention for the Riemann curvature tensor, $\kappa =8\pi G/c^4$: $G$ is the gravitational constant, whereas $G_{\mu \nu }$ is the Einstein tensor. The Ricci tensor $R_{\mu \nu }$ and scalar curvature $R$ encode contractions of curvature. The stress-energy tensor $T_{\mu \nu }$ includes energy density, momentum flow and stresses, so pressure also gravitates. A cosmological constant adds $\Lambda g_{\mu \nu }$ to the left-hand side. See Tong's Einstein equations and matter coupling.

For a concrete consequence, set $T_{\mu \nu }=0$ and $\Lambda =0$. The trace gives $R=0$, and substitution gives $R_{\mu \nu }=0$. Yet the exterior Schwarzschild spacetime still has the tidal forces computed under geodesic deviation: vanishing Ricci tensor does not imply vanishing Riemann curvature tensor. Geometry carries gravitational effects through empty space. Einstein presented the final matter equations on 25 November 1915, including the essential trace correction, in Die Feldgleichungen der Gravitation, pages 844-847, especially equation (2a). His symbols and sign conventions differ from modern ones; the displayed formula uses modern notation.

The stress-energy tensor describes local energy density, momentum density, and stress. Compatibility with the Einstein field equations requires covariant conservation ${\nabla }^{\mu }{T}_{\mu \nu }=0$.

An orthonormal frame $\{e_a\}$ satisfies $g(e_a,e_b)={\eta }_{ab}$. Any two such frames differ by a local Lorentz transformation.

At any spacetime point, normal coordinates make the metric equal to the Minkowski metric and all Christoffel symbols vanish at that point.

The Riemann curvature tensor measures the local path dependence of parallel transport. For a torsion-free connection in a coordinate basis, the conventiondefines its action through the failure of covariant derivatives to commute. Parallel transport around a small loop can consequently change a vector even when its starting and ending points coincide.

In general relativity, geodesic deviation connects curvature with the relative acceleration of neighbouring freely falling bodies: tidal stretching is a local physical effect. Nonzero connection coefficients alone do not establish curvature, since even a flat plane has nonzero coefficients in polar coordinates. Conversely, choosing coordinates that remove the connection at one point does not remove a nonzero curvature tensor. The overall curvature-sign convention must be preserved when comparing component formulas.^[ref]

A covariant derivative differentiates a tensor field while accounting for the changing local basis. For a vector field, an affine connection gives ${\nabla }_{\mu }{V}^{\nu }={\partial }_{\mu }{V}^{\nu }+{\Gamma }^{\nu }{}_{\mu \rho }{V}^{\rho }$. Each additional contravariant index contributes a plus connection term, and each covariant index contributes a minus term.

The correction makes the result a tensor, even under nonlinear changes of coordinates. For example, a constant Cartesian vector on a flat plane has varying components in polar coordinates; the connection terms cancel that coordinate variation. In general relativity, the Levi-Civita connection supplies the metric-compatible derivative. Parallel transport sets the derivative along a curve to zero, while the geodesic equation sets a freely falling body's tangent derivative to zero.^[ref]

Forthe only potentially nonzero Ricci component isThe vacuum equation is therefore the transverse Laplace equation.

A Killing vector field describes a direction in which geometry has a continuous symmetry: moving points together along its flow of a vector field preserves the metric tensor, including its pairing of tangent vectors. Such a symmetry gives a geodesic conserved quantity from a Killing vector, often a component of momentum or angular momentum. The direction may vary with position: rotation provides the basic example.

On a Riemannian manifold or Lorentzian manifold with metric tensor $g$, a smooth vector field $K$ is a Killing vector field when its Lie derivative of $g$ vanishes. If ${\Phi }_s$ is its local flow of a vector field, the equivalent statements areThe middle equality uses the pullback of a covariant tensor: the local flow maps are isometries on their domains. The last equality is the Killing equation, using the Levi-Civita connection. The infinitesimal condition holds throughout an open domain and integrates to exact local isometries; it is stronger than making the first derivatives of $g$ vanish at a single point. See Tong, section 4.3.1, equation (4.29).

For example, on the interval $0<x<1$ with Euclidean metric $d{x}^2$, $K={\partial }_x$ translates points by $x\mapsto x+s$. This is a local isometry, but a point reaches the end of the interval in finite flow parameter. A local flow of a vector field can be incomplete; see Killing fields on incomplete manifolds need not generate global isometries. The full-plane translation and rotation examples under Euclidean Killing vector field instead have flows for every real $s$.

Model theory studies mathematical first-order structures through the first-order formulas they satisfy. A theory specifies sentences, while a model of a first-order theory gives a domain and interpretations of the nonlogical symbols making those sentences true.

For example, dense total orders without endpoints have a common complete first-order theory, although a countable such order and the real line cannot be isomorphic. The Löwenheim-Skolem theorem and compactness theorem explain why first-order truth can leave cardinality and other global structure undetermined. Quantifier elimination, complete types, and elementary embeddings provide tools for describing exactly what a theory can distinguish.

Category theory studies mathematical structures through the morphisms between them. A category specifies which objects of a category and arrows are allowed; a functor translates these objects and arrows into another category; a natural transformation compares two such translations coherently. For example, the category of groups records both groups and group homomorphisms, so it remembers which maps respect multiplication.

The usefulness appears in constructions we already know. A Cartesian product combines two sets, while a direct product of groups combines two groups. Both satisfy the same universal property of a categorical product: any pair of maps into the factors comes from exactly one map into the product. This gives a reusable way to construct maps and establish isomorphisms, without depending on a particular choice of coordinates or element labels. In a partially ordered set regarded as a category, the corresponding categorical product is a greatest lower bound. The shared definition exposes a connection between pairing data and finding the best common bound.

Start with the computed examples under category, functor, natural transformation, and product in a category. Their arrow equations are practical tests: do both routes in a commutative diagram produce the same result? Later, the same language organizes tensor products, adjunctions, and the fundamental group of a topological space. Emily Riehl's textbook develops these basic notions before using them to compare constructions across different subjects. ^[ref]

In their 1945 paper, Samuel Eilenberg and Saunders Mac Lane set out the notions of category, functor, and natural transformation. The introduction uses the relation between a vector space and its dual space to ask which comparisons are natural, setting up a precise language for compatibility with maps. ^[ref]

A mathematical proof is a finite deductive argument from stated assumptions to a conclusion. It explains why the conclusion follows for every instance covered by its hypotheses. An informal mathematical proof can omit routine detail while making its assumptions and deductions clear; a formal proof expresses them in a specified formal language with specified inference rules. The distinction between assumptions, propositions, and deductions is introduced in Leighton and Rubinfeld, What is a proof?, sections 1–3.

Checking examples can suggest a pattern, but need not establish a universally quantified claim. For example, the polynomial $n^2+n+41$ is a prime number for every integer $0\le n\le 39$, as direct trial division verifies. At $n=40$ it equals $1681=41^2$, a composite number. Those first forty successes do not prove the claim for all natural numbers; one counterexample disproves it. The point is not to discard computation: it can find a conjecture, detect a mistake, or verify a finite exhaustive case analysis. The question is whether the cases checked cover the statement being claimed.

For a positive example, the sum of two even integers is even: writing them as $2a$ and $2b$ with $a,b\in \mathbb{Z}$ gives $2a+2b=2(a+b)$, and $a+b$ is an integer. This direct argument handles every allowed pair at once. A useful mathematical proof therefore does more than certify an answer: it exposes the mechanism and the exact assumptions that make the answer transferable.

Statistical learning theory studies mathematical guarantees and limitations for learning from data, including how a hypothesis class, loss function, and sample size affect generalization. It distinguishes a particular fitted rule's prediction error from the expected performance of the learning procedure across possible training data.