For metric $8|dz{|}^2/(1-|z{|}^2{)}^2$, these smooth fields satisfy the Bogomolny vortex equations with one simple zero at the origin. Their curvature coefficient is $F_{12}=4/(1+|z{|}^2{)}^2$, and their physical magnetic field is $(1-|z{|}^2{)}^2/[2(1+|z{|}^2{)}^2]=(1-|\Phi {|}^2)/2$. The magnetic flux is $2\pi$, and the critically coupled energy in the conformal-surface vortex square completion normalization is $\pi$.

Assume geodesic completeness. Minimize the distance to a target over a small compact geodesic sphere about the initial point. The distance splitting through a small geodesic sphere gives an initial direction satisfying $d(p,\gamma (t))+d(\gamma (t),q)=d(p,q)$. If this equality stops before the target, repeat the splitting at the last equality point. The resulting broken path minimizes, so its corner must smooth into the same continuing geodesic, a contradiction. This supplies a minimizing geodesic to every target. Every closed metric ball is then the image under the Riemannian exponential map of a closed tangent ball, hence compact. A Cauchy sequence has a convergent subsequence in such a ball and therefore converges, proving metric completeness without presupposing a global minimizing curve.

If the Riemannian exponential map at one point $p$ of a connected finite-dimensional Riemannian manifold is defined on all of $T_{p}M$, every point can be joined to $p$ by a minimizing geodesic. The distance splitting through a small geodesic sphere selects a radial direction attaining $d(p,q)=r+d(\gamma (r),q)$. If this equality stopped before the target, repeat the splitting at its last point. A broken minimizing path cannot have a corner, so the new segment continues the same geodesic, contradicting maximality. This is the minimizing-geodesic lemma used in the radial continuation proof of Hopf-Rinow.

// Target: riemannian-geometry.bigb

A simply connected closed three-dimensional manifold is orientable, since its orientation monodromy factors through its trivial fundamental group. Its first integral homology is the abelianization of that fundamental group and is zero. The universal coefficient theorem for cohomology gives $H^1(M;\mathbb{Z})=0$, and Poincare duality gives $H_2=0$, $H_3=\mathbb{Z}$. Connectedness gives $H_0=\mathbb{Z}$, and duality gives zero homology above dimension three. Thus it is an integral homology sphere, without needing any homeomorphism classification.

In the round real projective plane, project the great-circle arc $(\cos t,\sin t,0)$ for $0\le t\le \pi$. Its endpoint and velocity match after antipodal identification, so it is a closed geodesic. On the sphere its tangent and the constant vector $(0,0,1)$ are parallel. Endpoint identification fixes the returned tangent and reverses the returned transverse vector. Its parallel transport is therefore a reflection. Nonorientability permits determinant-minus-one holonomy.

An $SU(2)$ connection whose holonomy lies in a maximal circle, equivalently one preserving a splitting $E=L\oplus L^{-1}$. If its holonomy is the entire circle, parallel adjoint sections are exactly the one-real-dimensional diagonal Lie algebra. The off-diagonal real plane is the underlying real bundle of $L^2$ and has circle weight two.

// Destination: geometry-and-topology.bigb

Every nonzero subquotient of a holonomic Weyl algebra module over a characteristic-zero field has dimension $n$, by Bernstein inequality for Weyl algebra modules and dimension and multiplicity in a filtered exact sequence. Each strict step in a submodule chain consumes positive multiplicity of a filtered module. With a standard graded algebra normalization these multiplicities are integral. There are therefore at most $m(M)$ strict steps. The module is Artinian and Noetherian, and its module length is bounded by that multiplicity.

A holomorphic subbundle of a holomorphic vector bundle is a constant-rank vector subbundle which is locally spanned by a holomorphic linearly independent family of sections and whose inclusion is a holomorphic bundle map. It has an induced holomorphic vector bundle structure. A varying-rank coherent subsheaf is not generally a subbundle.

For a holomorphically trivial vector bundle, choose a global holomorphic bundle frame. Let the columns of $H_{\alpha }$ be the coordinates of that frame in the prescribed frame over $U_{\alpha }$. If fiber coordinates transform as $s_{\beta }=F_{\alpha \beta }{s}_{\alpha }$, then $H_{\beta }=F_{\alpha \beta }{H}_{\alpha }$. Each matrix is holomorphic and invertible, giving the displayed splitting. Conversely such matrices give agreeing global holomorphic sections forming a frame. Topological triviality alone does not supply this conclusion.

// Destination: electromagnetism.bigb

On a compact Riemann surface, the degree of a holomorphic vector bundle is the degree of a line bundle of its determinant line bundle. A Hermitian metric on a holomorphic vector bundle and its Chern connection express this degree by the trace-curvature integral. The value is independent of the metric because the normalized trace curvature represents the First Chern class.

A unitary representation of the fundamental group of a connected complex manifold gives a bundle $(\tilde{X}\times V)/{\pi }_{1}X$ with locally constant unitary transition matrices. Its induced flat unitary connection has $(0,1)$ part equal to its Dolbeault operator, so it is the Chern connection for the induced Hermitian metric. On a compact Riemann surface, zero curvature gives degree zero; the subbundle curvature inequality makes the bundle semistable.

The dual of the holomorphic tangent bundle is a holomorphic vector bundle with local frames $d{z}^1,\dots ,d{z}^n$. Coordinate changes have holomorphic invertible Jacobian matrices, so these frames have holomorphic transition functions. Its holomorphic sections are exactly the holomorphic one-forms.

A real connection on a vector bundle on $TM$ with $\nabla J=0$ restricts after complexification to a connection $D$ on the holomorphic tangent bundle. Conversely $D$ and its conjugate determine a unique such real connection. In holomorphic coordinates, compatibility with the holomorphic structure means $D_{{\partial }_{{\bar{z}}^i}}{\partial }_{z^j}=0$. Zero mixed torsion tensor says that this derivative equals ${\nabla }_{{\partial }_{z^j}}{\partial }_{{\bar{z}}^i}$; the two belong to opposite eigentypes, so both vanish. Reality proves the reverse implication. Preserving $J$ alone does not imply this condition.

Let $S_0$ be a commutative Noetherian ring, let a graded $S$ be generated over it by finitely many positive-degree homogeneous elements, and let $M$ be a finitely generated graded module. For an integer-valued function $\lambda$ additive on exact sequences of finitely generated $S_0$-modules, its graded-piece series has the displayed form. The exact sequence for multiplication by the last variable gives $(1-t^d)P_M=P_{M/xM}-t^d{P}_{(0{:}_{M}x)}$. Those two modules are finite over the ring with one fewer variable, and induction proves rationality. Finite graded pieces and the finite-generation assumptions cannot simply be omitted.

For $\mathrm{diag}(t,t^{-1})$, the monomial $x^{d-i}{y}^i$ has weight $d-2i$. Positive weights force divisibility by $x^{\lfloor d/2\rfloor +1}$. Conjugating the subgroup replaces $x$ by any nonzero linear factor. Thus a nonzero binary form is unstable exactly when a projective root has multiplicity greater than half its degree. Equality gives zero weight and does not imply instability.

The coordinates of a highest weight record its eigenvalues on a chosen basis of a Cartan subalgebra. They depend on that basis. Dynkin labels instead use the simple coroots; orthonormal Cartesian coordinates and Dynkin labels must be converted before applying a dimension or Casimir eigenvalue formula.

The high-pressure limit of the slow magnetosonic wave approaches an incompressible polarization in the plane of the background magnetic field and wavevector, perpendicular to the latter. It has the displayed leading dispersion relation, the same as the out-of-plane Alfvén wave. Magnetic tension is the restoring force, and pressure balances the compressive component of the magnetic force. In exactly incompressible homogeneous ideal magnetohydrodynamics it is one of the two transverse magnetic polarizations.

An admissible cut splits a four-dimensional cobordism as $W=W_1{\cup }_N{W}_2$, with $b_2^{+}(W_1),b_2^{+}(W_2)>0$, and with the Mayer-Vietoris connecting map $H^1(N;\mathbb{Z})\to H^2(W;\mathbb{Z})$ equal to zero. The last condition removes the ambiguity in gluing Spin-c structures on the two pieces. Positivity makes both infinity cobordism maps vanish, allowing passage through the Reduced Heegaard Floer homology of $N$. A useful cut is the boundary of a neighborhood of a positive-square embedded surface, chosen so that its complement still has a positive direction. The nonzero normal Euler number makes $H^1$ of the neighborhood surject onto $H^1$ of its boundary, proving the connecting-map condition.

For intersection generators $\mathbf{x},\mathbf{y}$, a Whitney disk is a map of a disk into the symmetric product, with its two boundary semicircles mapped to the two product tori and its endpoints mapped to $\mathbf{x},\mathbf{y}$. Its homotopy class belongs to ${\pi }_2(\mathbf{x},\mathbf{y})$. Projecting it to the Heegaard surface gives an integral domain. With the surface oriented, the convention is ${\partial }_{\alpha }D=\mathbf{y}-\mathbf{x}$. The obstruction to a disk is the class $ϵ(\mathbf{x},\mathbf{y})\in H_1(Y)$ obtained by joining the points along alpha arcs and returning along beta arcs. In the Heegaard Floer setup, its vanishing is equivalent to equality of the generators' Spin-c structures; for the rational homology sphere diagrams this is equivalent to ${\pi }_2(\mathbf{x},\mathbf{y})\ne \varnothing$.

For generators in the same Spin-c structure and a Whitney disk class $\psi$ joining them,The ambiguity is the divisibility of $c_1(\mathfrak{s})$ on $H_2(Y)$; a torsion determinant class gives an integer relative grading. Adding a sphere class changes $\mu$ by two and $n_z$ by one, so it leaves the difference unchanged. For a domain $D$, the index can be calculated as $\mu (D)=e(D)+n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D)$, where $e$ is Euler measure and each $n_{\mathbf{x}}$ is the sum of averages of the four adjacent region multiplicities at the selected intersections.