An isomorphism of Lie groups is a group isomorphism that is also a diffeomorphism. Both the homomorphism and its inverse preserve the smooth structures. Its identity differential is a Lie algebra isomorphism, but an isomorphism of Lie algebras determines a global Lie group isomorphism without extra period data only for connected simply connected groups.

A Lie group action on $M$ is proper if $(g,x)\mapsto (gx,x)$ from $G\times M$ to $M\times M$ is a proper map. A free proper smooth action has a smooth quotient manifold, with the quotient map a submersion and principal bundle. Proper actions with stabilizers can have orbit-type strata and singular quotients. These qualifications are necessary before treating an orbit space as a smooth phase space.

A Lie bracket is alternating when $[x,x]=0$ for every $x$. This condition is imposed in every characteristic of a field. Bilinearity gives $0=[x+y,x+y]=[x,y]+[y,x]$, hence antisymmetry of a Lie bracket. Requiring alternation directly keeps the definition correct in characteristic two.

If the matrix logarithm identifies a neighborhood of the identity in a Matrix Lie group with a neighborhood of zero in its tangent subspace, the logarithm of the displayed group commutator belongs to that subspace. Divide it by $t^2$ and take the limit. Finite-dimensional linear subspaces are closed, proving closure under the matrix commutator without attempting to give a nonsmooth square-root reparametrization a velocity.

For a connected simply connected source Lie group, every Lie algebra homomorphism integrates uniquely to a Lie group homomorphism. Along a path $\gamma$ from the identity, transport its left logarithmic velocity by $\theta$ and solve the corresponding left-invariant differential equation in the target. The Maurer-Cartan equation and preservation of brackets make the endpoint invariant under fixed-endpoint homotopies. Simple connectedness removes path dependence, and path concatenation proves multiplicativity.

For trivial real coefficients, a Lie algebra two-cocycle is an alternating bilinear form $\kappa :\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$ satisfyingOn the vector space $\mathfrak{g}\oplus \mathbb{R}$, define $[(X,a),(Y,b)]=([X,Y],\kappa (X,Y))$. The second summand is central, and the displayed identity is exactly the remaining condition for the Jacobi identity. Thus $\kappa$ defines a central extension of a Lie algebra. Replacing the chosen linear map splitting the quotient by $X\mapsto (X,b(X))$ changes $\kappa$ by $-b([X,Y])$; this is a coboundary in Lie algebra cohomology. For the abelian plane, $\kappa ((x,y),(x^{\prime },y^{\prime }))=x{y}^{\prime }-y{x}^{\prime }$ gives the Heisenberg Lie algebra. The constant discrepancy in a moment-map equivariance obstruction is another such Lie algebra two-cocycle.^[ref]

On a compact boundaryless Riemannian spin manifold with $R\ge 0$, integrating the Lichnerowicz spinor-square formula against a harmonic spinor gives $\int (|\nabla \psi {|}^2+R|\psi {|}^2/4)=0$. Positivity forces the spinor to be parallel. Its norm is constant on each connected component, so a nontrivial solution is nowhere zero there. Strictly positive scalar curvature at any point on such a component excludes nonzero harmonic spinors.

For a metric-compatible connection with torsion tensor $T^{\rho }{}_{\mu \nu }={\Gamma }_{\mu \nu }^{\rho }-{\Gamma }_{\nu \mu }^{\rho }$, expand the square using Clifford multiplication and the full affine second derivative. Its antisymmetric part is spin curvature minus $T^{\rho }{}_{\mu \nu }{\nabla }_{\rho }$. This gives the displayed formula. The curvature need not have the torsion-free pair symmetry or first Bianchi identity, so its Clifford contraction need not reduce to a scalar $-R/4$. General torsion produces additional first-order and nonscalar curvature couplings.

The four-gamma product decomposes into fully antisymmetric, two-gamma and scalar parts. The first part vanishes on contraction by the first Bianchi identity, and the two-gamma parts vanish because they contract symmetric Ricci curvature against antisymmetric gamma matrices. The scalar contractions are $-g^{ac}{g}^{bd}{R}_{abcd}+g^{ad}{g}^{bc}{R}_{abcd}=-2R$. Here $R=g^{ac}{g}^{bd}{R}_{abcd}$ fixes the curvature-slot convention.

// Target: second-fundamental-form.bigb

For $\alpha >0$, the Lévy process with characteristic function $\mathbb{E}{e}^{iu{X}_t}=(1+u^2/{\alpha }^2{)}^{-t}$ has zero Gaussian coefficient, zero uncompensated drift, and the displayed Lévy measure. Indeed, the symmetric jump integral is $2{\int }_0^{\infty }(\cos (uy)-1)e^{-\alpha y}dy/y$. Differentiating in $u$ gives $-2u/({\alpha }^2+u^2)$, and integrating back from zero gives $-\log (1+u^2/{\alpha }^2)$. The time-one law is the Laplace distribution of rate $\alpha$. The jump activity is infinite, since $K$ has infinite mass near zero, but the jump variation is finite because $\int (1\wedge |y|)K(dy)<\infty$.

If a Lévy measure satisfies ${\int }_{|y|\le 1}|y|K(dy)<\infty$, the jump integral in the Lévy–Khintchine formula can be written without compensation: $\mathbb{E}{e}^{iu{X}_t}=e^{t\psi (u)}$, with the displayed exponent. Its integral is absolutely convergent because $|e^{iuy}-1|\le |u||y|$ near zero and is at most $2$ elsewhere. If the compensated formula uses drift $\gamma$, the uncompensated drift is $b=\gamma -{\int }_{|y|\le 1}yK(dy)$. The Gaussian coefficient is still ${\sigma }^2\ge 0$.

In the convention $⟨0|J^{\mu }|P(p)⟩=-i\sqrt{2}{F}_P{p}^{\mu }$, the Fermi interaction gives $\mathcal{M}=G_F{F}_P{p}_{\mu }{\bar{u}}_{\ell }{\gamma }^{\mu }(1-{\gamma }_5)v_{\nu }$. With a massless antineutrino, the external Dirac equations reduce this to $G_F{F}_P{m}_{\ell }{\bar{u}}_{\ell }(1-{\gamma }_5)v_{\nu }$. The spin sum is $8G_F^2{F}_P^2{m}_{\ell }^{2}k\cdot q=4G_F^2{F}_P^2{m}_{\ell }^2(M_P^2-m_{\ell }^2)$. The two-body Lorentz-invariant phase space has integrated value $(1-m_{\ell }^2/M_P^2)/(8\pi )$, and the initial flux factor is $1/(2M_P)$, giving the displayed width. If the decay constant is instead defined without the square root of two, $f_P=\sqrt{2}{F}_P$ and the denominator is $8\pi$. A weak mixing coefficient multiplies the amplitude and its modulus squared multiplies the width.

At tree level with universal weak charged current coupling and massless neutrinos, the leptonic pseudoscalar decay width gives the displayed ratio. The pion decay constant, Fermi constant and common quark mixing coefficient cancel. The charged-lepton mass factor is helicity suppression: a spin-zero parent cannot emit the two massless helicities selected by the chiral weak interaction. The ratio tests this suppression and lepton universality; scalar or pseudoscalar four-fermion interactions can modify it. Radiative corrections must be included for a precision comparison.

Lorentz covariance allows only a multiple of $p^{\mu }$ for a vacuum-to-spin-zero one-particle current. In its rest frame only the temporal component can remain. A parity-even temporal vector current between a parity-even vacuum and parity-odd particle changes sign, forcing that component to vanish. A temporal axial current is parity-odd and is allowed. This explains why the pion decay constant enters the axial, rather than vector, part of the weak charged current.

For small categories $\mathcal{C},\mathcal{D}$, a functor $F:\mathcal{C}\to \mathcal{D}$ and $Q:\mathcal{C}\to \mathbf{Set}$, choose an infinite cardinal number $\kappa$ bounding the number of morphisms of $\mathcal{D}$ and the total number of elements of $Q$. Given a natural transformation $\eta :Q\to GF$, generate a subfunctor $G^{\prime }\subseteq G$ from all ${\eta }_c(x)$ by applying every outgoing $\mathcal{D}$-morphism. Each $G^{\prime }(d)$ has at most $\kappa$ elements. Label bounded values by subsets of $\kappa$; only a set of such functors and transformations exists. These factorizations give the solution-set condition for precomposition, whose left adjoint is the left Kan extension.

For small categories and a set-valued functor $Q$, each displayed colimit exists in the Category of sets. A representative $(c,u:Fc\to d,x\in Qc)$ maps under a natural transformation $\eta :Q\to GF$ to $G(u){\eta }_c(x)$. The colimit identifications make this well-defined, proving the adjunction between left Kan extension and precomposition.

For a left-exact reflective subcategory of a Cartesian closed category, an ambient exponential $B^A$ is already fixed by the reflector when $A,B$ are fixed. Indeed maps into $B^A$ transpose to maps from $X\times A$ into $B$, and reflection gives $L(X\times A)\cong LX\times A$. Precomposition with the reflection unit is therefore bijective for maps into $B^A$, characterizing a fixed object.

Declare a global left-invariant, respectively right-invariant, frame parallel to obtain two flat affine connections on a Lie group. For left-invariant vector fields, their derivatives are as displayed and their torsion tensors are $-[X,Y]$ and $+[X,Y]$. The tensors vanish on Abelian groups; two genuinely torsionful connections are not asserted there.

The standard four-torus has trivial complex tangent bundle, so its symplectic canonical class is zero. The symplectic adjunction formula gives $F^2=2g-2$ for a smooth genus-$g$ fiber of a symplectic Lefschetz pencil. Two generic fibers meet once positively at each base point and nowhere else, giving the displayed count.

Let $X$ be a smooth projective surface over the complex numbers, and let $H$ be an ample divisor. For sufficiently large integers $m$, a general pencil in $|mH|$ is a Lefschetz pencil. Its smooth fiber represents $mH$, so the arithmetic adjunction formula on a smooth surface givesIf $m=4k$, then $m^2{H}^2/2=8k^2{H}^2$ and $m{K}_X\cdot H/2=2k{K}_X\cdot H$ are both even integers. Therefore $g$ is odd. Positivity is needed to obtain a generic Lefschetz pencil; divisibility by four is the separate parity argument. The genericity assertion here uses complex algebraic geometry, rather than an unqualified positive-characteristic version of the complex Morse local model.^[ref]