If $\mathfrak{a}$ is an abelian ideal of a Lie algebra, then ${\mathrm{ad}}_x{\mathrm{ad}}_y$ maps $\mathfrak{g}$ into $\mathfrak{a}$ and vanishes on $\mathfrak{a}$ whenever $x\in \mathfrak{a}$. Its trace is zero, so $\kappa (\mathfrak{a},\mathfrak{g})=0$. Every nonzero solvable ideal has a last nonzero term in its derived series of a Lie algebra, which is such an abelian ideal. Consequently a nondegenerate Killing form forces the solvable radical to vanish.

A finite-dimensional real Lie algebra has negative-definite Killing form exactly when it is compact semisimple. In the compact direction, average an inner product over a compact group; all adjoint maps are skew, so $\kappa (X,X)=-\Vert {\mathrm{ad}}_X{\Vert }^2$, and semisimplicity removes the center. Conversely, negative-definiteness gives the positive invariant metric $-\kappa$, embeds the faithful adjoint algebra in an orthogonal algebra, and nondegeneracy gives semisimplicity. This is an algebraic compactness criterion, not a claim that every global group with an Abelian compact-type algebra is compact.

A probability measure on the circle invariant under $T_2$ and $T_3$ and ergodic for their jointly generated semigroup is Lebesgue measure if either transformation has positive Kolmogorov-Sinai entropy. Without joint ergodicity, a mixture of Lebesgue measure and an atomic invariant measure shows why positive entropy alone is insufficient.

For the structure sheaf of a complex manifold, local holomorphic logarithms give the short exact sequence of sheavesThe connecting map $H^1(M,{\mathcal{O}}_M^{\ast })\to H^2(M,\mathbb{Z})$ sends a holomorphic line bundle to its First Chern class. In particular, it is onto when $H^2(M,{\mathcal{O}}_M)=0$.

This complex has the sheaves ${\Omega }_M^p$ of holomorphic differential forms and differential $\partial$. By the holomorphic Poincaré lemma, it resolves the complex constant sheaf.

A holomorphic $(p,0)$ form is annihilated by $\bar{\partial }$ and, by bidegree, by ${\bar{\partial }}^{\ast }$. The Kähler Laplacian identity makes it harmonic for $d$, hence $d$-closed on a compact Kähler manifold.

On a compact complex surface, the claim also holds without a Kähler metric. For a holomorphic one-form $\alpha$, let $\beta =d\alpha$. It is a closed holomorphic two-form, and the Stokes theorem gives $\int \beta \wedge \bar{\beta }=\int d(\alpha \wedge \bar{\beta })=0$. Positivity of $\beta \wedge \bar{\beta }$ forces $\beta =0$. Functions are constant on each connected component, and holomorphic two-forms are closed by dimension.

A section $\omega$ of the canonical bundle of a Riemann surface, locally $\omega =\varphi (z)dz$ with holomorphic $\varphi$. A nonzero form on a compact genus-$g$ surface has total zero order $2g-2$. Its integral supplies flat coordinates away from zeros.

For a nonvanishing local $m$-differential $\varphi (z)d{z}^m$, choose $w=\int {\varphi }^{1/m}dz$. Then the differential is $d{w}^m$. Different root choices give $w\mapsto \zeta w+b$, ${\zeta }^m=1$. For $m=1,2$ the permitted linear parts are translations or signs, producing translation surfaces and half-translation surfaces.

If a local holomorphic $r$-differential is $z^{m}u(z)d{z}^r$, where $u(0)\ne 0$, its zero order is $m$. The order is unchanged by a holomorphic coordinate change with nonzero derivative. For a holomorphic one-form the cone angle is $2\pi (m+1)$, and for a holomorphic quadratic differential it is $(m+2)\pi$.

On a connected compact Kähler manifold of complex dimension two, the intersection form restricted to real $(1,1)$ cohomology has signature $(1,h^{1,1}-1)$. The Kähler class spans a positive line, and its orthogonal complement consists of primitive classes, which are negative definite. Thus a totally isotropic subspace of real $(1,1)$ cohomology has dimension at most one.

Every point $(P_x,P_y,P_z)$ can be reached from the identity by a smooth curve tangent to the Heisenberg horizontal distribution. Put $a=2(P_z-P_x{P}_y/2+P_x)$ and chooseTheir integrals give $x(1)=P_x$, $y(1)=P_y$, and $z(1)={\int }_0^{1}x(t)\beta (t)dt=P_x{P}_y/2-P_x+a/2=P_z$. The third coordinate measures the effect of the noncommuting horizontal directions; integrability would instead confine horizontal curves to a two-dimensional leaf.

For an elliptic curve over a number field, its nonnegative canonical height of an elliptic curve has finite bounded subsets by the Northcott theorem and its bounded difference from half the naive logarithmic height. By the Weak Mordell-Weil theorem, choose representatives $R_i$ modulo $2E(K)$ and put $H={\max }_i\hat{h}(R_i)$. If $P=2Q+R_i$, the Cauchy-Schwarz inequality for the height pairing gives $\hat{h}(Q)\le (\hat{h}(P)+H)/2$. Iterating brings every point into the finite set of height at most $H+1$. This set and the $R_i$ generate $E(K)$, proving the Mordell-Weil theorem without assuming finite generation when constructing the height.

For a closed smooth embedding $i:N\hookrightarrow M$ of codimension $r$ whose normal bundle is oriented over a coefficient ring $R$, a tubular neighborhood, excision and the Thom isomorphism theorem identifyThe relative-to-absolute map defines $i_{!}:H^j(N;R)\to H^{j+r}(M;R)$. For closed oriented $M,N$, it is characterized bywith compatible orientations. It obeys $i_{!}(b⌣i^{\ast }c)=i_{!}b⌣c$. If $j:M\setminus N\hookrightarrow M$ and $\delta :H^q(M\setminus N;R)\to H^{q+1-r}(N;R)$ is the connecting map followed by the inverse Thom isomorphism, then, over ${\mathbb{F}}_2$,These identities follow from naturality of the relative cup product and show how the exact sequence determines multiplication in complements. Over ${\mathbb{F}}_2$ every real normal bundle has the required orientation.

For an invertible probability measure-preserving system with a finite or countable finite-entropy generating measurable partition $\xi$, $h_{\mu }(T)=h_{\mu }(T,\xi )$. For a noninvertible system the same conclusion holds for a finite-entropy one-sided generator. The finite-entropy hypothesis is part of this form of the theorem.

A one-sided generator is a measurable partition $\xi$ with $\sigma ({\bigvee }_{j\ge 0}{T}^{-j}\xi )=\mathcal{B}$ modulo null sets. Equivalently, every measurable set can be approximated in measure by unions of atoms of finite forward-name blocks. A one-sided coordinate partition generates a one-sided Bernoulli shift; for a nontrivial base distribution, it does not generate the two-sided version, whose negative coordinates are independent of the nonnegative ones.

For odd $p$, a one-dimensional formal group law over ${\mathbb{Z}}_p$ has a formal logarithm ${\log }_F(T)=T+{\sum }_{j\ge 2}{c}_j{T}^j/j$ with $c_j\in {\mathbb{Z}}_p$. This follows by integrating its integral invariant differential. For $t\in p{\mathbb{Z}}_p\setminus \{0\}$, $v_p(c_j{t}^j/j)>v_p(t)$ for $j\ge 2$. Hence the series converges and $v_p({\log }_F(t))=v_p(t)$, so its kernel is zero. As it is a homomorphism into a characteristic-zero additive group, there is no nonzero torsion. In particular, at an odd prime of good reduction over $\mathbb{Q}$, reduction injects the whole rational torsion subgroup, including its $p$-primary part.

Suppose bounded $V\ge 0$ is bounded below by a positive constant on some nonempty open interval, and $f\ge C>0$. The Feynman-Kac formula with a bounded potential gives $u(t,x)\ge C{\mathbb{E}}_x{e}^{{\int }_0^{t}V(B_s)ds}$. Infinite occupation time of one-dimensional Brownian motion makes that integral tend to infinity almost surely. The monotone convergence theorem then gives $u(t,x)\to \infty$ for every fixed $x$. Thus no globally bounded classical solution exists under these hypotheses, although boundedness on every finite time slab is compatible with the conclusion.

For a pole off the real axis, substitution $s=2t$ into the Faddeeva function integral gives $G(a)=i\pi w(a/2)$ when $\mathrm{Im}a>0$, and $G(a)=-i\pi w(-a/2)$ when $\mathrm{Im}a<0$. The difference of boundary values is the residue theorem jump $2\pi i{e}^{-a^2/4}$. In particular, for a lower-half-plane pole, $G(a)+2\pi i{e}^{-a^2/4}=i\pi w(a/2)$, which packages a crossed residue and a nearby saddle into one smooth expression.

On $y^2=x^3+ax+b$ in characteristic of a field other than two, take the elementary symmetric polynomials $s_i$ of $x(P),x(Q),x(P+Q)$. If the line through $P,Q$ is $y=\lambda x+\mu$, its third intersection is $-(P+Q)$, so coefficient comparison gives $s_1={\lambda }^2$, $s_2=a-2\lambda \mu$, $s_3={\mu }^2-b$. Hence $(s_2-a{)}^2=4s_1(s_3+b)$. This is an identity of rational functions; tangent cases follow by specialization, and coordinates at infinity are handled through the projective addition map.