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For a homogeneous incompressible flow, strict geostrophic balance means that the Coriolis acceleration is exactly balanced by the pressure gradient, after conservative body-force potentials have been absorbed into the pressure per unit density $\pi$. It neglects material acceleration rather than merely assuming that its size is small. With constant vertical rotation, ${\pi }_z=0$ and taking the curl gives ${\mathbf{u}}_z=0$, the Taylor–Proudman theorem. Nonzero cross-contour motion over sloping rigid boundaries requires small departures from this strict limit.

If a map on a normed vector space satisfies $\Vert T(x)\Vert \le q\Vert x\Vert$ for every $x$, with $0\le q<1$, induction gives $\Vert T^n(x)\Vert \le q^n\Vert x\Vert$. The geometric series then proves absolute convergence of its orbit series whenever the ambient normed vector space is complete. The scalar map $T(x)=\sin (qx)$ satisfies the bound because $|\sin u|\le |u|$.

For $|r|<1$, let $S_N$ contain the first $N$ terms, with $S_0=0$. The remainder after those terms isThis is the exact absolute error. A convenient bound replaces the denominator by $1-|r|$, giving $|R_N|\le |a||r{|}^N/(1-|r|)$ through the triangle inequality. The exact denominator is preferable when the sign or complex phase of $r$ is known.

For $a>0$ and $0<r<1$, the omitted fraction of the entire sum is $R_N/(a/(1-r))=r^N$. Capturing at least $99$ percent therefore means $r^N\le 0.01$. Taking logarithms and applying the ceiling function gives the smallest integer $N$:This gives $7,21,90,$ and $459$ terms for $r=0.5,0.8,0.95,$ and $0.99$, respectively. The difference is a concrete reason to care about the common ratio, even when all four series converge.

An absolute target has a different scale. For $a=1$, $r=0.8$, and required absolute error at most $0.01$, the total sum is $5$ andThe first acceptable term count is $N=28$: the remainder is approximately $0.0096714$, while at $N=27$ it is approximately $0.0120893$. Counting terms and checking the tail avoids an ambiguous instruction to stop when a term merely looks small.

Prequantization realizes a symplectic manifold using a Hermitian line bundle with connection whose curvature is $i\omega /\hslash$ in the convention ${\iota }_{X_f}\omega =df$. Its existence requires integral symplectic periods. On sections, the Kostant-Souriau prequantum operator represents every classical observable and its Poisson bracket, but the resulting states depend on the full phase space. A polarization in geometric quantization removes this excess of variables.

A polarization is an involutive complex rank-half distribution on which the symplectic form vanishes. Polarized prequantum sections satisfy ${\nabla }_{X}s=0$ for $X\in \mathcal{F}$. A real polarization is tangent to a Lagrangian foliation; a complex Kähler polarization uses the antiholomorphic tangent spaces. Only observables whose Hamiltonian flows preserve the chosen distribution act directly on the polarized space. Singular leaves, holonomy and changes of polarization are genuine quantization difficulties.

Let a uniformly expanding interval map have cylinders ${\Delta }_w$ whose maximum length at level $n$ is at most $C{\lambda }^{-n}$, with $\lambda >1$. Suppose an invariant probability density $h$ is Lipschitz continuous with constant $L$ and satisfies $h\ge h_{\ast }>0$. Its cylinder probabilities are $\nu ([w])={\int }_{{\Delta }_w}hdx$. Define the piecewise constant logarithmic averageThe average of $h$ over ${\Delta }_w$ differs from $h(x)$ by at most $L\mathrm{diam}({\Delta }_w)$. Since the logarithm has derivative at most $1/h_{\ast }$ on this range,Thus the cylinder data recover the logarithm of the density uniformly at a geometric rate.

Conversely, take consistent positive cylinder probabilities and suppose these $g_n$ converge uniformly to a finite function $g$. Put $h=e^g$. If $\Vert g_n-g{\Vert }_{\infty }\le {\delta }_n\to 0$, integrating over every level-$n$ cylinder compares its $h$-mass with its assigned probability within factors $e^{\pm {\delta }_n}$. Subdivide any fixed cylinder into these finer cylinders and sum: consistency gives the same multiplicative bounds for the fixed cylinder. Letting $n\to \infty$ proves ${\int }_{{\Delta }_w}hdx=\nu ([w])$.

Shrinking cylinders determine the Borel probability measure, so the geometric image of the symbolic law is exactly $h(x)dx$. Use a consistent convention at branch endpoints; bounded limiting cylinder densities imply zero mass at individual endpoints. If the symbolic law is shift invariant, the coding relation transfers that invariance to the interval map. This converse is much stronger than inspecting a density ratio along just one itinerary.

A geometric morphism $p:\mathcal{D}\to \mathcal{F}$ is localic when $\mathcal{D}$ is, over $\mathcal{F}$, the topos of sheaves on an internal locale of $\mathcal{F}$. Equivalently, subobjects of the objects $p^{\ast }X$, with $X\in \mathcal{F}$, form a separating family in $\mathcal{D}$.

A geometric morphism $h$ is hyperconnected when its inverse image functor of a geometric morphism is full and faithful and its essential image is closed under subobjects. It then identifies all subobjects of $h^{\ast }X$ with subobjects of $X$, and its image is closed under quotients as well, by descending the kernel-pair relation.

For a geometric morphism $f:\mathcal{E}\to \mathcal{F}$, the displayed internal frame represents $X\mapsto {\mathrm{Sub}}_{\mathcal{E}}(f^{\ast }X)$. The localic part is ${\mathrm{Sh}}_{\mathcal{F}}(L)\to \mathcal{F}$. Its basic open families pull back to the subobjects of $f^{\ast }X$ they represent; gluing these gives a full faithful, subobject-closed inverse image, hence the hyperconnected geometric morphism part. The represented subobject functor recovers the internal frame, proving uniqueness up to equivalence over $\mathcal{F}$.

If a bounded linear operator from a Banach space onto a normed vector space admits the displayed lift for every $y$, then its target is complete. Given a Cauchy sequence, choose a subsequence with summable successive differences. Lift those differences to a summable sequence in the source, and use its convergent sum to produce a limit of the target subsequence. The original Cauchy sequence has the same limit.

For a vector field $Z$, define $\beta (v,w)=⟨{\nabla }_{v}Z,w⟩+⟨{\nabla }_{w}Z,v⟩$. Its diagonal evaluation is $2⟨{\nabla }_{v}Z,v⟩$. Along a geodesic this equals $2d⟨Z,\dot{\gamma }⟩/dt$, so it is invisible to closed-geodesic integrals. If a fibrewise linear function $u(x,v)={\eta }_x(v)$ solves $Xu=\beta (v,v)$, then $Z={\eta }^{♯}/2$ gives this convention.

On a closed oriented Riemannian surface this integral identity follows from the canonical commutators and integration by parts against Liouville volume of a surface geodesic flow. If $VX\phi =-2H\phi$ and $K<0$, substitution gives $\Vert X\phi {\Vert }^2+5\Vert H\phi {\Vert }^2+\int (-K)(V\phi {)}^2=0$. Each term is nonnegative, forcing all three derivatives to vanish. For $\phi =V^{2}u+u$, the fibre equation then makes $u$ a constant plus a fibrewise linear function, supplying the potential symmetric 2-tensor kernel.

The displayed one-form on the unit round sphere has zero integral on every closed geodesic. Indeed, the antipodal map pulls it back to its negative, and the second half of an oriented great circle is the antipodal image of the first. Nevertheless $d\theta =2x_{3}d{x}_3\wedge d{x}_1$ is nonzero on an open set, so the one-form is not a closed differential form and cannot be an exact differential form. Thus a curvature-free extension of one-form injectivity fails.

For a surface metric $Ed{u}^2+2Fdudv+Gd{v}^2$, all coordinate curves are unparametrized geodesics precisely when the displayed identities hold. Parametrize $u=c$ by arc length, so $\dot{v}=G^{-1/2}$, and substitute into the transverse Euler-Lagrange equation for energy of a curve. This gives the first identity; $v=d$ gives the second. These identities concern geodesic images and do not assert that the coordinate variable itself is an affine parameter.

For $p$ in a Riemannian manifold and $v$ sufficiently close to zero in $T_{p}M$, let ${\gamma }_v$ be the geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The exponential map isEquivalently, ${\exp }_p(tv)={\gamma }_v(t)$ wherever both sides are defined.

A spacelike geodesic has positive tangent norm in signature $(-+++)$ and satisfies the geodesic equation with an affine parameter. Metric compatibility preserves that norm along the curve. In Minkowski spacetime these are straight lines whose spatial tangent is larger in magnitude than the time component; their two ends approach spacelike infinity in a conformal completion.

For a surface metric $(U(u)+V(v))(d{u}^2+d{v}^2)$ with $U+V>0$, an affinely parametrized geodesic has constant energy $E=(U+V)({\dot{u}}^2+{\dot{v}}^2)/2$. Put $p=(U+V)\dot{u}$. The Euler-Lagrange equations give $\dot{p}=E{U}^{\prime }/(U+V)$. Thus $K=p^2-2EU$ has derivative zero, proving the displayed additional quadratic integral. This separation structure is a geometric example of Liouville integrability.

The velocity of a geodesic can be extended to one smooth ambient vector field on a sufficiently short neighbourhood of any parameter value. If its initial velocity is zero, uniqueness of the geodesic equation makes it locally constant, and the zero field works. Otherwise one coordinate has nonzero velocity; the inverse function theorem supplies coordinates straightening the small embedded arc to $(t,0,\dots ,0)$. The coordinate field $\partial /\partial t$ is the required extension. Shrinking the interval is essential: arbitrary fields along self-intersecting curves need not have one ambient extension.

Two points along a geodesic are conjugate when a nonzero Jacobi field along it vanishes at both points. This detects singularity of the geodesic exponential map and governs loss of local length minimization.

For a continuous map of nonzero mapping degree between closed connected oriented surfaces, rational cohomology injectivity of a nonzero-degree map gives an injection on $H^1$. Since the first Betti number of an oriented surface of genus $g$ is $2g$, the source genus is at least the target genus.