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 be the coordinates of that frame in the prescribed frame over . If fiber coordinates transform as , then . 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 with locally constant unitary transition matrices. Its induced flat unitary connection has 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 . 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 with restricts after complexification to a connection on the holomorphic tangent bundle. Conversely and its conjugate determine a unique such real connection. In holomorphic coordinates, compatibility with the holomorphic structure means . Zero mixed torsion tensor says that this derivative equals ; the two belong to opposite eigentypes, so both vanish. Reality proves the reverse implication. Preserving alone does not imply this condition.
Let be a commutative Noetherian ring, let a graded be generated over it by finitely many positive-degree homogeneous elements, and let be a finitely generated graded module. For an integer-valued function additive on exact sequences of finitely generated -modules, its graded-piece series has the displayed form. The exact sequence for multiplication by the last variable gives . 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 , the monomial has weight . Positive weights force divisibility by . Conjugating the subgroup replaces 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 , with , and with the Mayer-Vietoris connecting map 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 . 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 of the neighborhood surject onto of its boundary, proving the connecting-map condition.
For intersection generators , 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 . Its homotopy class belongs to . Projecting it to the Heegaard surface gives an integral domain. With the surface oriented, the convention is . The obstruction to a disk is the class 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 .
For generators in the same Spin-c structure and a Whitney disk class joining them,The ambiguity is the divisibility of on ; a torsion determinant class gives an integer relative grading. Adding a sphere class changes by two and by one, so it leaves the difference unchanged. For a domain , the index can be calculated as , where is Euler measure and each is the sum of averages of the four adjacent region multiplicities at the selected intersections.
An embedded positive domain which is a bigon or rectangle, has convex alternating alpha and beta corners, and contains no other selected coordinate in its interior, has Maslov index one. A bigon has Euler measure and two corner averages ; a rectangle has Euler measure zero and four corner averages . For the usual regular complex structure, its holomorphic disk count modulo translation is . For a bigon the Riemann map to the strip is unique up to translation. For a rectangle, the tautological correspondence represents the disk in the symmetric product by a branched double cover of the strip onto the rectangular domain: its conformal modulus fixes the branch parameter, and the only remaining freedom is translation. The resulting isolated regular disk gives one point in the oriented moduli space. The differential coefficient is , so a basepoint-free domain contributes a unit.
Over , holomorphic disk moduli spaces in a Heegaard Floer chain complex must be oriented compatibly with gluing: the orientation of a concatenated disk class is the product determinant-line orientation of its two factors. Boundary orientations of compactified one-dimensional moduli spaces then make broken disks cancel, yielding . Isolated regular disks contribute or . Changing the chosen orientation at an intersection generator changes the corresponding row and column signs of the differential and yields an isomorphic integral chain complex. Cobordism triangle counts require compatible orientation choices as well; an overall orientation ambiguity accounts for the usual sign ambiguity of the closed invariant.
In normal coordinates , write the Riemannian volume form as and let . With the heat operator , the Gaussian times a power-series amplitude has its singular terms cancelled by the displayed radial equations for . The initial coefficient is , andMultiplying the equation by gives the derivative of . Smoothness forces the integration constant to vanish. The integral on a fixed compact interval proves smoothness across the diagonal by induction; it yields . A truncated heat parametrix therefore has residual .
The normalized pullback is a unitary map from of the quotient onto the deck-invariant subspace of of the cover. Local isometry and the sheet-counting integration formula preserve the Dirichlet quadratic form. Therefore the Friedrichs extension and its heat semigroup intertwine with this map. A finite sum of lifted Riemannian heat kernels gives the quotient kernel; independence of lifts follows by reindexing the deck group. This formulation avoids a uniqueness claim for unrestricted heat-equation solutions on a noncompact or incomplete manifold.
On a closed two-dimensional Riemannian manifold, the first three integrated scalar heat invariants are , , and , where is Gaussian curvature. If its scalar spectrum agrees with that of a hyperbolic surface of area , these integrals are . Expanding the displayed square then forces everywhere. Thus constant negative curvature is spectrally determined in this setting, although the global hyperbolic metric need not be determined up to isometry.
For the positive scalar Laplace-Beltrami operator on a closed manifold with a Riemannian metric, the coefficient of in the normalized heat trace is the displayed integral. The local coefficient additionally has a scalar-curvature divergence term whose integral vanishes. In dimension two, , , , giving . Together with volume and total Gaussian curvature, it determines the variance of curvature and detects constant curvature.
For positive real numbers and nonnegative weights with , the weighted harmonic mean isEqual weights give the harmonic mean, while a zero weight excludes that input. For travel at positive constant speeds over positive leg lengths , the weights are fractions of total distance:For at and at , the weights are , elapsed time is hours, and the average speed is . Weighting the speeds by their time fractions instead gives the same result as a weighted arithmetic mean: . The two weight systems describe the same journey in different coordinates and must not be interchanged. [ref].
There are unlisted articles, also show them or only show them.