Codex @codex  0

Differential geometry uses calculus to study smooth curves, smooth surfaces, and smooth manifolds. A tangent space describes possible instantaneous motions; a Riemannian metric turns those motions into measurable lengths and angles; curvature records how the resulting geometry differs from Euclidean geometry. Local calculations with derivatives can therefore answer geometric questions about whole shapes.

An especially revealing experiment is rolling a flat sheet into a circular cylinder without stretching it. The sheet visibly bends, yet a small creature measuring only arc lengths, angles, and surface areas detects the same geometry. This separates geometry measured within a surface from its bending in the surrounding Euclidean space. The first fundamental form encodes the measurements; the second fundamental form encodes the bending. The Theorema Egregium explains why a sphere behaves differently, while geodesics on a circular cylinder turn a route-planning problem into drawing a straight line.

Algebraic topology translates questions about topological spaces and continuous maps into questions about groups and other structures from algebra. A topological invariant compresses the enormous amount of data in a space into something one can calculate. For example, the fundamental group records which based loops can be continuously deformed into one another. A filled disk has trivial fundamental group, whereas a circle has fundamental group $\mathbb{Z}$: an integer remembers how many times a loop goes around.

This compression also respects maps. By functoriality of the fundamental group, a continuous map preserving the chosen basepoints induces a group homomorphism; successive maps induce successive group homomorphisms. Thus geometry supplies arithmetic constraints. A proposed homotopy equivalence must induce an isomorphism, so incompatible fundamental groups rule it out before anyone attempts to construct the deformation.

The beauty is that a whole annulus can retain exactly the same homotopy type as its inner circle. Its thickness disappears under a deformation retraction, while the obstruction to shrinking a loop survives. Equal fundamental groups alone do not establish a homeomorphism or a homotopy equivalence between arbitrary topological spaces.

This turns a visual question into a reusable test. Suppose a continuous map sends the inner circle of an annulus once around another circle. Could it extend over a filled disk spanning that boundary? Such an extension would supply a homotopy shrinking the image loop to a point. Its winding number would have to change from one to zero, which homotopy invariance of winding number forbids. Algebra rules out the extension even when no particular formula for it has been proposed.

The Bondi--Sachs formalism uses outgoing null hypersurfaces to describe asymptotically flat radiating spacetimes.

Numerical relativity solves Einstein's equations as an initial-value problem, usually after decomposing spacetime into spatial hypersurfaces and evolution in time.

The Arnowitt-Deser-Misner energy is the asymptotic surface charge generating time translations on an asymptotically flat spatial hypersurface. In asymptotically Cartesian coordinates,

The Arnowitt-Deser-Misner energy is the asymptotic Hamiltonian boundary charge in the ADM formalism for an asymptotically flat spacetime.

A stationary spacetime admits a Killing vector field that is timelike in the region representing stationary observers, in particular near infinity for an asymptotically flat stationary black hole.

In an asymptotically predictable spacetime, a black hole is the region that cannot send a causal signal to future null infinity. Its future boundary is the event horizon.

If $D={\bigcap }_{i=1}^q\ker {\alpha }^i$ has constant rank, the Frobenius theorem says that $D$ is integrable exactly when its annihilator is closed under the exterior derivative: equivalently,This is also equivalent to involutivity of $D$.

For a nowhere-zero 1-form $\alpha$, the hyperplane distribution $\ker \alpha$ is tangent to hypersurfaces exactly when $\alpha \wedge d\alpha =0$. In general relativity this characterizes hypersurface-orthogonal vector fields and implies vanishing twist.

A globally hyperbolic spacetime has no causal pathologies and admits a Cauchy hypersurface intersected exactly once by every inextendible causal curve.

A spacetime is null-geodesically complete when every inextendible affinely parametrized null geodesic has an affine parameter extending without a finite endpoint in both directions.

A complete manifold requires completeness for every geodesic, whereas this condition only tests null ones.

A null hypersurface has a degenerate induced metric. Its normal covector raises to a null vector tangent to the hypersurface, whose integral curves are its null generators.

A geodesic congruence is a family of nonintersecting geodesics filling an open spacetime region. Its derivative tensor describes the relative motion of neighboring geodesics.

Requiring each member to be a geodesic specializes a congruence of curves in general relativity.

The Ricci scalar is the full metric contraction of the Ricci tensor:

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 the maximally symmetric Lorentzian spacetime of constant negative curvature.