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A positive linear functional on the space of continuous compactly supported functions of a locally compact Hausdorff space has a unique representing positive Radon measure. Positivity gives boundedness on fixed compact supports: choose a compactly supported continuous cutoff $p\ge 1$ on the support $K$, then $|I(f)|\le \Vert f{\Vert }_{\infty }I(p)$ for real $f$ supported in $K$. The resulting measure is finite on compact sets but need not have finite total mass.

A (pseudo-)Riemannian manifold is geodesically complete when every inextendible affinely parametrized geodesic is defined for all real parameter values. In the positive-definite case, the Hopf-Rinow theorem identifies this with metric completeness. Null-only completeness in spacetime is a separate weaker condition.

On a connected Riemannian manifold of dimension at least three, pointwise independence of sectional curvature from the tangent two-plane forces the curvature to be constant. The contracted Bianchi identity differentiates the relation $\mathrm{Ric}=(n-1)Kg$ and gives $(n-2)dK=0$.

A geodesic polygon is a closed polygonal curve whose sides are geodesic segments. A geodesic triangle has three sides. Surface angle excess is governed by the Gauss-Bonnet theorem.

A Riemannian metric satisfying the displayed equation for a vector field $X$ and constant $\lambda$. A gradient Ricci soliton has $X=\nabla f$; it is steady when $\lambda =0$.

A minimal submanifold has vanishing mean curvature vector, equivalently zero first variation of volume for compactly supported normal variations. A minimal surface is its two-dimensional case. A calibrated submanifold satisfies the stronger property of minimizing volume in its relative homology class.

// Target: general-relativity.bigb

For $1<\sigma \le 2$, the three-four-one inequality for Euler products gives $|1/\zeta (\sigma +it)|\le \zeta (\sigma {)}^{3/4}|\zeta (\sigma +2it){|}^{1/4}$. For $|t|\ge 1$, the fractional-part continuation formula for the Riemann zeta function, truncated at $H\asymp 2+|t|$, gives ${\zeta }^{(j)}(\sigma +it)=O_j({\log }^{j+1}(2+|t|))$. Thus the reciprocal bound even holds with exponent $1/4$ instead of $1/2$ in this range. On bounded-height compact sets, three-four-one product proof of zeta boundary nonvanishing and removal of the reciprocal's zero at the pole give uniform bounds. Differentiating $1/\zeta$ repeatedly yieldsThe range restriction matters: $1/\zeta (\sigma )\to 1$ as $\sigma \to \infty$, so the first displayed bound cannot hold uniformly for all $\sigma >1$. A global version replaces $(\sigma -1{)}^{-3/4}$ with $1+(\sigma -1{)}^{-3/4}$.

Absolute convergence of $1/\zeta (s)=\sum \mu (n)n^{-s}$ for $\mathrm{Re}s>1$ gives $|1/\zeta (\sigma +it)|\le \zeta (\sigma )=O((\sigma -1{)}^{-1})$ as $\sigma \downarrow 1$. If $\zeta$ is holomorphic at $1+i{t}_0$, a zero of order $m$ there would give reciprocal growth of order $(\sigma -1{)}^{-m}$ on the horizontal approach. Thus $m\le 1$. This proves a multiplicity restriction, not existence of such a zero or a zero-free-line theorem.

For a compact Riemann surface with a symplectic basis $a_i,b_i$ of its first homology group, cutting along that basis and integrating a primitive givesfor closed smooth one-forms. For a holomorphic differential form $\omega$ and a differential of the third kind $\eta$, the corresponding boundary calculation givesThe paths and representatives of the cycles are chosen in the same cut surface. Normalized holomorphic differential forms consequently have a symmetric period matrix with positive definite imaginary part.

A meromorphic one-form on a compact Riemann surface with only simple poles is a differential of the third kind. Prescribed residues can be realized exactly when their sum is zero: in the residue exact sequence for $K_C(S)$, the obstruction in $H^1(C,K_C)\cong \mathbb{C}$ is the sum of residues. Subtracting a holomorphic differential form normalizes its $a$-periods. Integer residues and periods in $2\pi i\mathbb{Z}$ make the exponential of its integral a single-valued meromorphic function with those orders of zeros and poles.

Integration of holomorphic differential forms along cycles embeds the first homology group as a full Euclidean lattice in $H^0(C,K_C{)}^{\vee }$. Fixing a base point, sum the path integrals to the points of an effective complex analytic divisor. Different path choices differ by that lattice. The Abel theorem for divisors identifies the fibres with linear equivalence of divisors.

A Riemann surface which is compact as a topological space. For a connected one of genus $g$, the Riemann-Roch theorem gives $g$ independent holomorphic differential forms. The Riemann bilinear relations for a compact surface make their normalized periods a full lattice in ${\mathbb{C}}^g$, and the Abel-Jacobi map of a compact Riemann surface turns this lattice into a complex torus encoding divisor classes.

A noncompact Riemann surface is hyperbolic in potential theory when its Green-function exhaustion has a finite limit, equivalently when it admits a positive Green function on a Riemann surface. This definition does not assume a disk uniformization. The unit disk is an example, since $-\log |z|$ is its positive Green function with pole zero. For simply connected surfaces this is equivalent to being conformally equivalent to the disk. For general surfaces, potential-theoretic hyperbolicity should not be conflated with the condition that the universal cover is the disk: some surfaces with disk universal cover have no positive Green function.

A meromorphic differential is specified in each holomorphic chart by a meromorphic coefficient $f(z)$, with the coefficients transforming by the coordinate derivative so that the one-form agrees on overlaps. The derivative of a meromorphic function gives such a differential, and the ratio of two nonzero differentials is a meromorphic function. Orders of zeros and poles are independent of chart because coordinate derivatives are nonvanishing holomorphic factors. On a compact connected surface the sum of orders of a nonzero differential is $2g-2$, allowing its divisor to determine the genus.

Given any positive $k(n)\to \infty$, choose increasing integers $n_j$ with $k(n_j)\ge j{2}^j$. The absolutely uniformly convergent series $g(t)={\sum }_j{2}^{-j}{e}^{i{n}_{j}t}$ is continuous, and its Fourier coefficient at $n_j$ is exactly $2^{-j}$. The displayed unbounded limsup follows, although every individual continuous function still has coefficients tending to zero by the Riemann-Lebesgue lemma.

For a continuous function $f:[a,b]\to \mathbb{R}$ and an integrator $g$ of bounded variation, the Riemann-Stieltjes integral is the common limit, as the mesh of a partition tends to zero, ofUnlike an ordinary Riemann sum, it weights each sample by the change in $g$, rather than by the interval length. Uniform continuity of $f$ and the finite total variation of a function bound make the tagged sums Cauchy: passing to a common partition refinement changes a sum by at most the modulus of continuity of $f$ at its mesh times the total variation of a function of $g$.

If $g(x)=x$, this is the ordinary Riemann integral. If $g$ has a single unit jump at an interior point $c$ and is otherwise constant, the value is $f(c)$: all its weight concentrates at that point. For a continuous nondecreasing $g$, the associated Lebesgue-Stieltjes measure, with no atom at $a$, satisfies $\mu ((u,v])=g(v)-g(u)$ and gives the same integral. More general discontinuous integrators require attention to endpoint and one-sided conventions when identifying their signed measures.

Vector-valued integrators are treated componentwise. In particular, a continuous path of bounded variation permits the iterated Riemann-Stieltjes integrals defining its path signature.

Changing the value of a bounded Riemann-integrable function at one point preserves its Riemann integral. The difference is supported at that point: partitions isolating it in intervals of arbitrarily small total length make both Darboux sums tend to zero. Thus a nonnegative single-point spike can have zero integral without vanishing everywhere. Its indefinite integral is identically zero, so the fundamental theorem of calculus derivative identity can fail at the changed point despite differentiability of the primitive.

A small-angle cone rotating above a stationary plate imposes approximately uniform shear rate, since local gap and wall speed both scale with radius. Integrating the uniform shear stress with moment arm r over the plate gives the displayed torque. The leading approximation neglects edge effects and inertia.

For an incompressible flow of constant mass density $\rho$ and kinematic viscosity $\nu$, Reynolds averaging of the Navier-Stokes equation, without a body force, givesHere $\mathbf{u}=\mathbf{U}+{\mathbf{u}}^{\prime }$ and $\overline{{\mathbf{u}}^{\prime }}=0$, with averaging assumed to commute with the displayed derivatives. The mean acceleration ${\partial }_t{U}_i$ vanishes only when the averaged velocity field is stationary; taking an average does not by itself remove it.

To derive the extra term, average the conservative momentum flux: $\overline{u_i{u}_j}=U_i{U}_j+\overline{u_i^{\prime }{u}_j^{\prime }}$. Mean incompressibility turns the divergence of the first product into $U_j{\partial }_j{U}_i$. The second product is the Reynolds stress divided by mass density, using the positive-covariance convention. It describes transport of mean momentum by fluctuating velocities. A turbulence closure is needed because that covariance is an additional unknown, rather than a specified molecular transport coefficient.

An eddy viscosity parametrizes turbulent momentum transport by $\overline{u^{\prime }{v}^{\prime }}=-{\nu }_T{U}_y$. The mixing length model gives ${\nu }_T=l^2|U_y|\ge 0$. This coefficient depends on the mean flow and position; it is distinct from molecular kinematic viscosity. It describes momentum transport rather than molecular energy dissipation.