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Every simply connected Riemann surface is biholomorphic to the Riemann sphere, the complex plane, or the unit disc.

If $\pi :S\to R$ is a covering of a Riemann surface, compose every chart on an evenly covered open set with each local inverse sheet of $\pi$. The resulting transition maps are those of $R$, so they define a unique complex structure making $\pi$ a local biholomorphism.

A germ at $z\in D$ is an equivalence class $[f{]}_z$ of holomorphic functions defined near $z$, where two representatives are equivalent when they agree on some neighbourhood of $z$.

It is the holomorphic-function case of a germ.

A map between Riemann surfaces is holomorphic when its expression in every pair of local complex coordinates is a holomorphic function.

For $2\le p<\infty$,Thus every function in $L^2\cap L^{\infty }$ belongs to every $L^p$ between them.

If a radial function on ${\mathbb{R}}^n$ behaves like $r^a$ near zero and like $r^{-b}$ at infinity, then its $q$th power is locally integrable at zero exactly when $aq>-n$ and integrable at infinity exactly when $bq>n$. This follows from the radial measure factor $r^{n-1}dr$.

A smooth compressive simple wave forms a shock when characteristics first intersect. If characteristics are parametrized by their emission time $\tau$, the first shock occurs at the minimum positive time for which $\partial x(\tau ,t)/\partial \tau =0$.

In a right-moving simple wave entering gas initially at rest with sound speed $c_0$, the other Riemann invariant is constant, soThe isentropic relationconverts the velocity disturbance into its nonlinear pressure disturbance.

For a nonnegative unbounded function, one truncation convention sets $f_r=\min (f,r)$ and asks that every $f_r$ be Riemann integrable and that $\int f_r$ have a finite limit as $r\to \infty$.

A lower Darboux sum uses the infimum of a function on each partition interval, while an upper sum uses the supremum.

A bounded function is Riemann integrable exactly when for every $\epsilon >0$ some partition $P$ satisfies $U(f,P)-L(f,P)<\epsilon$.

The total variation of a function $f:[a,b]\to \mathbb{R}$ measures its accumulated rises and falls, rather than just its change between the endpoints. For each partition of an interval $\Pi =(a=t_0<\cdots <t_n=b)$, add the absolute values of its sampled increments. Then take the supremum over all such partitions of an interval:The function has bounded variation when $V_a^b(f)<\infty$. This definition uses its value at every point of the interval, including both endpoints.^[ref]

For a nondecreasing function, all increments are nonnegative and the sum telescopes to $f(b)-f(a)$. More generally, the triangle inequality gives $|f(b)-f(a)|\le V_a^b(f)$. Equality can fail dramatically when the function changes direction: an object can travel a long distance and finish near its starting position. Here we measure movement of the function's value; arc length of its function graph also includes horizontal movement.

Every Riemann-integrable function on a compact interval can be approximated in $L^1$ by continuous functions. Consequently their integrals converge uniformly over all subintervals.

For a tagged partition $a=x_0<\cdots <x_n=b$ with $t_i\in [x_{i-1},x_i]$, the corresponding Riemann sum isIts limits over increasingly fine partitions define the Riemann integral.

In dimension greater than two, if sectional curvature is pointwise independent of the two-plane, the contracted Bianchi identity forces that curvature to be constant on each connected component.

The positive-definite specialization is Schur's lemma.

The Ricci tensor and scalar satisfy ${\nabla }^{\alpha }{R}_{\alpha \beta }=\frac{1}{2}{\nabla }_{\beta }R$.

The algebraic symmetries of the Riemann curvature tensor leave only one independent curvature component in two dimensions and forceConsequently the Einstein tensor vanishes identically.

A Drude response has a single relaxation pole, $G_R(\omega )={\sigma }_0/(1-i\omega {\tau }_0)$. Causality places the pole in the lower half plane when ${\tau }_0>0$.

The spectral function records the weighted energy differences connected by an operator. With the convention $G_R=-i\theta ⟨[A,A]⟩$, it is proportional to $-\mathrm{Im}{G}_R$.