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Use algebraic Chern classes in the Chow ring of a nonsingular algebraic variety $Y$. For a vector bundle $E$ of rank $r$, the classes $c_i(E)\in {\mathrm{CH}}^i(Y)$ are invariant under bundle isomorphism, have $c_0(E)=1$, vanish for $i>r$, and commute with pullback. Their Total Chern class is $c(E)={\sum }_i{c}_i(E)$. The Whitney sum formula for Chern classes applies to every short exact sequence $0\to E^{\prime }\to E\to E^{\prime \prime }\to 0$:Normalization for a line bundle is $c({\mathcal{O}}_Y(D))=1+[D]$, where $[D]$ is the divisor class. In particular, $c_1(L\otimes M)=c_1(L)+c_1(M)$, $c_1(L^{-1})=-c_1(L)$, and the trivial bundle has total class one. These properties determine the classes: pull back to the iterated projective bundle where $E$ has a filtration by line bundles. The projective bundle formula for Chow groups makes this pullback injective, and Whitney multiplication expresses $c_i(E)$ as the elementary symmetric polynomial of the first classes of the line factors. This also gives $c_i(E^{\ast })=(-1{)}^i{c}_i(E)$.

Let $X$ be the smooth degree-$d$ projective hypersurface and set $h=c_1({\mathcal{O}}_X(1))$. The algebraic cotangent bundle has rank $n-1$. Restriction of the cotangent Euler sequence in homogeneous coordinates remains exact, because its terms and quotient are locally free. Thusand Whitney multiplication givesThe defining equation has degree $d$, so the conormal sheaf is ${\mathcal{O}}_X(-d)$. Smoothness makes its differential map injective with locally free quotient, giving the Conormal exact sequence for Kähler differentialsA second application of the Whitney sum formula for Chern classes yieldsThe inverse is the finite geometric series in the graded Chow ring: terms in codimension greater than $n-1$ vanish. Expanding gives every requested class,For example,when the corresponding ranks and codimensions exist. The first expression agrees with the canonical bundle of a smooth projective hypersurface $\det {\Omega }_X^1={\mathcal{O}}_X(d-n-1)$. For a plane curve, $\mathrm{deg}{c}_1=d(d-3)=2g-2$, matching Q4. For a quartic surface in ${\mathbb{P}}^3$, $c_1=0$ and $c_2=6h^2$, whose degree is $24$. These checks also confirm the sign in the cotangent, rather than tangent, formula.

Let $X$ be a separated Noetherian scheme over a field $k$ and $\mathcal{F}$ a coherent sheaf. Coherent cohomology studies the groups $H^q(X,\mathcal{F})$: global sections in degree zero and the obstructions to gluing local sections in higher degrees. Coherence means finite generation locally, and on a Noetherian scheme it is preserved by kernels, cokernels and extensions. This makes coherent sheaves suitable for exact-sequence calculations.

For an ordered open cover $\mathcal{U}=(U_i)$, the Čech cochain groups arewith differentialEach double omission occurs twice with opposite signs, so ${\delta }^2=0$. The Čech cohomology of the cover is ${\check{H}}^q=\ker \delta /\mathrm{im}\delta$, and taking the direct limit over refinements defines cover-independent Čech cohomology. In degree zero the sheaf gluing axiom identifies it with $\Gamma (X,\mathcal{F})$.

The comparison with derived-functor sheaf cohomology is particularly effective algebraically. Positive-degree cohomology of a quasi-coherent sheaf on an affine scheme vanishes. A separated scheme has affine intersections of affine opens, so a finite affine cover is acyclic for $\mathcal{F}$. Its Čech cochain complex consequently computes $H^q(X,\mathcal{F})$ directly. For a short exact sequence of sheaves, sections over each affine intersection are exact on quasi-coherent sheaves. The resulting short exact sequence of Čech complexes gives the long exact sequence in sheaf cohomology. In particular, the connecting map measures the failure of a global section of the quotient to lift globally.

Several central projective results make the theory finite and computable. On a projective variety over $k$, every coherent cohomology group is finite-dimensional. The Grothendieck vanishing theorem gives $H^q(X,\mathcal{F})=0$ for $q>\dim X$. If ${\mathcal{O}}_X(1)$ is ample, Serre vanishing gives $H^{q>0}(X,\mathcal{F}(m))=0$ for sufficiently large $m$, and these large twists are globally generated sheaves. On projective space a coherent sheaf has a finite twisting resolution of a coherent sheaf on projective space: finite sums of $\mathcal{O}(a)$ resolve it. The explicit line-bundle cohomology below and the long exact sequences prove finiteness and positive-twist vanishing. For a closed embedding $i:X\hookrightarrow {\mathbb{P}}^n$, $i_{\ast }\mathcal{F}$ is coherent and $H^q(X,\mathcal{F})=H^q({\mathbb{P}}^n,i_{\ast }\mathcal{F})$, reducing these results to projective space. A sufficiently positive twist of a finite presentation by sums of $\mathcal{O}(-a)$ also proves global generation.

The Euler characteristic of a coherent sheafis therefore defined, and the long exact sequence makes it additive. With a fixed projective embedding, $\chi (X,\mathcal{F}(m))$ is its Hilbert polynomial, and for sufficiently large $m$ it equals $h^0(X,\mathcal{F}(m))$. On a smooth projective variety of dimension $r$, Serre duality gives, for a vector bundle $E$,For an arbitrary coherent sheaf the correct counterpart is $H^q(X,\mathcal{F}{)}^{\ast }\cong {\mathrm{Ext}}_X^{r-q}(\mathcal{F},{\omega }_X)$; replacing this Ext group indiscriminately by the cohomology of the ordinary sheaf dual would lose local extension information.

We now calculate the basic examples. For $n\ge 1$, use the standard cover $U_i=D_{+}(x_i)$ of ${\mathbb{P}}^n$. Sections of $\mathcal{O}(m)$ on an intersection are degree-$m$ Laurent monomials in which negative exponents are allowed only for coordinates inverted on that intersection. Decompose the Čech cochain complex by exponent vectors $(a_0,\dots ,a_n)$ with sum $m$. If no exponent is negative, the corresponding simplex complex contributes only in degree zero. If all exponents are negative, the monomial occurs only on the full intersection and contributes only in degree $n$. When the negative-index set is nonempty and proper, inserting any index outside it gives a contracting homotopy, so the monomial complex is acyclic. Hence the cohomology of twisting sheaves on projective space isIn the top group a basis consists of the Laurent monomials with every exponent at most $-1$. Writing $a_i=-1-b_i$ reduces the count to nonnegative solutions of $\sum b_i=-m-n-1$, giving the displayed binomial coefficient. In particular, all intermediate cohomology vanishes, and ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$ agrees with Serre duality. For ${\mathbb{P}}^0$ separately, every twist is trivial and $H^0=k$.

On the projective line this givesThus positive and negative degrees produce quite different groups, while their alternating difference is a single polynomial. The twisted cubic is isomorphic to ${\mathbb{P}}^1$ and its hyperplane bundle corresponds to ${\mathcal{O}}_{{\mathbb{P}}^1}(3)$. Consequently $h^0(X,{\mathcal{O}}_X(m))=3m+1$ for $m\ge 0$, agreeing with Q2, and $h^1(X,{\mathcal{O}}_X(m))=\max (-3m-1,0)$.

For a smooth plane curve $C$ of degree $d\ge 1$, the ideal-sheaf sequence of a projective hypersurface isThe long exact sequence in sheaf cohomology and the preceding formula give $H^0(C,{\mathcal{O}}_C)=k$ and $H^1(C,{\mathcal{O}}_C)\cong H^2({\mathbb{P}}^2,\mathcal{O}(-d))$. ThereforeFor $d=1,2$ this is zero; for a plane cubic it is one. Twisting the same exact sequence gives $\chi (C,{\mathcal{O}}_C(m))=dm+1-g$. These examples illustrate how Čech cohomology, vanishing, Serre duality and exact sequences convert local equations into global numerical invariants.

The usual field norm $N_{L/K}$ is defined for a finite field extension. The printed hypothesis “algebraic” alone allows an infinite extension, for which this determinant norm is not defined. We therefore assume $[L:K]<\infty$. For an arbitrary algebraic extension the meaningful elementwise statement is instead $N_{K(b)/K}(b)\in A$, and we prove this as well.

First a unique factorization domain is an integrally closed domain. Indeed, write an integral fraction as $u/v$ with relatively prime numerator and denominator. A monic integral equation, after multiplication by $v^r$, shows that $v$ divides $u^r$. Unique factorization forces $v$ to be a unit.

Let $m_b(T)=T^r+a_{r-1}{T}^{r-1}+\cdots +a_0\in K[T]$ be the monic minimal polynomial of $b$. Because $b$ is an integral element over $A$, it satisfies a monic polynomial $f\in A[T]$, and $m_b$ divides $f$ in $K[T]$. Every root of $m_b$ in an algebraic closure is therefore integral over $A$. This assertion includes repeated roots when the extension is inseparable. The coefficients of $m_b$ are elementary symmetric expressions in these roots. The integral elements form a ring: finitely many integral elements generate a finite $A$-module algebra, and the determinant trick shows every element of that algebra is integral. Thus all $a_i$ are integral over $A$. Since $a_i\in K$ and $A$ is integrally closed, $a_i\in A$.

Multiplication by $b$ on $K(b)$, in the basis $1,b,\dots ,b^{r-1}$, has the companion matrix of $m_b$, whose determinant is $(-1{)}^r{a}_0$. If $e=[L:K(b)]$, take a $K(b)$-basis of $L$. Multiplication by $b$ preserves each summand in the resulting direct sum of $e$ copies of $K(b)$, so its determinant over $K$ isIn particular $N_{K(b)/K}(b)=(-1{)}^r{a}_0\in A$. The argument requires no separability; in fact it holds for every integrally closed domain, not just a unique factorization domain.

For the connection with dimension, let $B$ be the coordinate ring of an irreducible affine variety. By Noether normalization there is a finite integral inclusion $A=k[t_1,\dots ,t_d]\subset B$, with $d=\dim B$. Their fraction fields form a finite extension. For any $0\ne b\in B$, its minimal-polynomial constant term $a_0$ is nonzero and lies in $A$. Its equation givesThus a nonzero function on the normalized variety produces a nonzero function on the base that vanishes wherever $b$ vanishes. Up to sign, the field norm is a power of this $a_0$, so it too belongs to $bB\cap A$.

Under the finite morphism to ${\mathbb{A}}^d$, the image of $V_B(b)$ is therefore contained in the proper Zariski-closed set $V_A(a_0)$ of dimension at most $d-1$. Finite morphisms preserve the dimension of a closed subvariety and its image. Applying this to a nonzero element of the ideal of any proper closed subvariety proves proper closed subvarieties have strictly smaller dimension. This is the useful role of the norm argument: nonzero equations upstairs give nonzero equations downstairs under a finite normalization.

Use the graded ring homomorphismwhere source degree $m$ maps to target degree $3m$. The homogeneous ideal of the twisted cubic is $\ker \phi$: a homogeneous polynomial vanishes on the parametrized curve precisely when its substitution vanishes identically. Here $k$ is infinite, being an algebraically closed field, so vanishing at all parameter points implies the polynomial identity.

Let $J$ be generated byTheir substitutions vanish, hence $J\subseteq \ker \phi$. To prove the reverse inclusion as an equality of ideals, we need more than a set-theoretic description of the zero locus.

Modulo $J$, replace $x_1^2$ by $x_0{x}_2$, $x_2^2$ by $x_1{x}_3$, and $x_1{x}_2$ by $x_0{x}_3$. Each replacement strictly decreases the sum of the exponents of $x_1$ and $x_2$, so the reduction terminates. It shows that the degree-$m$ quotient is spanned by the standard monomials of the twisted cubicFor $m\ge 1$ there are $(m+1)+m+m=3m+1$ of these. For $m=0$ the sole monomial is $1$. Their substituted images have $t$-exponents respectively $3b$, $3b+1$ and $3b+2$, so they are exactly the distinct monomialsThese are linearly independent. Consequently any degree-$m$ polynomial in $\ker \phi$, after reduction modulo $J$, has all its reduced coefficients zero. It belongs to $J$. Applying this to each homogeneous component givesThis also proves that the quotient embeds in the domain $k[s,t]$, so the displayed ideal is prime; no radicalization or saturation correction is concealed in the argument. Its image is the third Veronese subring ${⨁}_{m\ge 0}k[s,t{]}_{3m}$, and the calculation gives the Hilbert polynomial $3m+1$, consistent with degree three and genus zero.

We work over an algebraically closed field $k$, so the assertion concerns a geometric projective line. Let $V=k^4$ and let $W$ be the vector space of homogeneous cubic forms on $V$. It has dimension $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$, so the parameter space of cubic equations up to nonzero scalar is $\mathbb{P}(W)={\mathbb{P}}^{19}$. The Grassmannian $G=\mathrm{Gr}(2,V)$ parametrizes lines in ${\mathbb{P}}^3$. Its standard graph charts are four-dimensional affine spaces, so its dimension is four. The Plücker embedding realizes it as a closed quadric in ${\mathbb{P}}^5$, proving projectivity. The general linear group ${\mathrm{GL}}_4$ acts transitively on two-planes and is an irreducible open subset of the affine space of matrices; its orbit map proves that $G$ is irreducible.

Consider the incidence varietyRestriction of a cubic to a fixed line is surjective onto the four-dimensional space of binary cubic forms: after choosing coordinates with that line given by $x_2=x_3=0$, extend any binary cubic using $x_0,x_1$ alone. Thus its kernel has dimension sixteen. On each Grassmannian chart these restriction maps depend algebraically on the line parameters and have constant rank four. Their kernels form a rank-sixteen vector bundle, and $\mathcal{I}$ is its projective bundle, with fibre ${\mathbb{P}}^{15}$. In particular, $\mathcal{I}$ is irreducible of dimension $4+15=19$. It is closed in the product, since restriction vanishes precisely when four polynomial coefficients vanish.

The projection $\pi :\mathcal{I}\to \mathbb{P}(W)$ is a projective morphism, hence its image is closed. It remains to prove that this image has dimension nineteen. Equal dimensions alone would not suffice: we must control a fibre locally.

TakeIn the open Grassmannian chart around ${\ell }_0$, every line is uniquely a graphRestriction of $F_0$ to that line isThe four monomials are linearly independent over every field. Hence the only line in this chart lying on $F_0$ has $a_0=a_1=b_0=b_1=0$. The fibre ${\pi }^{-1}([F_0])$ therefore has local dimension zero at $({\ell }_0,[F_0])$. We do not need $F_0$ to define a smooth surface, nor do we claim its entire fibre is finite.

The fiber dimension theorem, in its local dimension inequality, givesThus the closed image has dimension at least nineteen. A proper closed subset of the irreducible ${\mathbb{P}}^{19}$ has smaller dimension, so $\pi (\mathcal{I})={\mathbb{P}}^{19}$. Every cubic equation consequently has a line in its zero set. In particular, every nonsingular cubic surface contains a line. The proof in fact establishes existence for arbitrary cubic surfaces over an algebraically closed field; it does not assert a rational line over an arbitrary nonclosed ground field.

Let $D$ be the Levi-Civita connection of the Riemannian metric. We use the curvature sign specified in the question, the negative of the usual vector-bundle curvature convention in Question 4. Extend $D$ to the covariant exterior derivative on tangent-bundle-valued forms and define the curvature two-form by $R=-D^2$. In a local frame with connection matrix $A$, it is the endomorphism-valued two-form $-(dA+A\wedge A)$. The transformation law for $A$ makes this transform by conjugation, so it is an intrinsic tensor. It measures the infinitesimal failure of parallel derivatives to commute, with this chosen sign.

For a tangent-bundle-valued one-form $\eta$, the covariant exterior derivative isApply this to the one-form $DZ$, for which $(DZ)(Y)=D_{Y}Z$. This calculation proves, rather than just renames, the operator expressionIt is linear over smooth functions in $Z$: the extra first- and second-derivative terms in $R(X,Y)(fZ)$ cancel by $[X,Y]f=X(Yf)-Y(Xf)$. The connection's linearity in its vector-field argument similarly shows linearity over smooth functions in $X$ and $Y$. Thus $R(X,Y)$ is an endomorphism of the tangent bundle, not a differential operator on its sections, and $R$ is a well-defined endomorphism-valued two-form.

The torsion-free identity is $D_{X}Y-D_{Y}X=[X,Y]$. Expand the cyclic curvature sum and use this identity to obtainThe last equality is the Jacobi identity for the Lie bracket of vector fields. This proves the first Bianchi identity

Lower the output index with the metric and define the Riemann curvature tensor byThis is a covariant four-tensor, because the curvature endomorphism is tensorial in all three inputs. Skew-symmetry of the curvature two-form immediately gives $R_{ji,kl}=-R_{ij,kl}$.

Metric compatibility gives the second skew-symmetry. Apply the scalar operator $[X,Y]-XY+YX$, which is zero, to $g(Z,W)$. Expanding each derivative using $Dg=0$ cancels the mixed products of first derivatives and leavesThus $R_{ij,lk}=-R_{ij,kl}$.

For a direct proof of pair interchange, choose the geodesic coordinates of Question 5 at an arbitrary point $p$. There $\Gamma (p)=0$ and $\partial g(p)=0$. Write $D_{{\partial }_i}{\partial }_k={\Gamma }^a{}_{ik}{\partial }_a$. The operator formula gives, at $p$,The Levi-Civita formula is $g_{al}{\Gamma }^a{}_{jk}=\frac{1}{2}({\partial }_j{g}_{kl}+{\partial }_k{g}_{jl}-{\partial }_l{g}_{jk})$. Differentiating at $p$, the products involving first derivatives of the metric and Christoffel coefficients vanish. The terms ${\partial }_i{\partial }_j{g}_{kl}$ cancel, leavingHere a comma denotes ordinary coordinate derivatives at the normal-coordinate center. Swapping $(i,j)$ with $(k,l)$ leaves this expression unchanged, by symmetry of $g$ and commutation of its second partial derivatives. This is the pair symmetry from the normal-coordinate curvature formula. Since $p$ was arbitrary and the equality is tensorial, it holds in every frame. We have proved all the requested symmetries:Changing the overall curvature sign would change the displayed coordinate formula's sign, but would not change the first Bianchi identity or any of these algebraic symmetries.

Fix $p$ in a Riemannian manifold $(M,g)$. The preliminary results used are the existence of the Levi-Civita connection, which is metric-compatible and torsion-free; local existence, uniqueness and smooth dependence for the geodesic ordinary differential equation; and the inverse function theorem. The local geodesic equation isso these ordinary differential equation results apply to the initial data $(p,v)$. On an open neighborhood of $0\in T_{p}M$, the solution ${\gamma }_v$ with ${\gamma }_v(0)=p$, ${\dot{\gamma }}_v(0)=v$ is defined up to time one, and the exponential map is ${\exp }_p(v)={\gamma }_v(1)$.

The geodesic equation is invariant under affine rescaling of the parameter. Uniqueness therefore gives ${\gamma }_v(t)={\exp }_p(tv)$ whenever these expressions are defined. In particularThe inverse function theorem now makes ${\exp }_p$ a diffeomorphism from a sufficiently small ball about zero in $T_{p}M$ onto a neighborhood of $p$. Choose a $g_p$-orthonormal basis $(e_i)$ and define coordinates byThese are geodesic coordinates, also called normal coordinates. They exist near every point. Radial coordinate rays $x^i(t)=t{v}^i$ are affinely parametrized geodesics issuing from $p$, and $g_{ij}(p)={\delta }_{ij}$. Substitution of all such rays into the geodesic equation at $t=0$ gives ${\Gamma }^i{}_{jk}(p)v^j{v}^k=0$ for all $v$. Since the connection is torsion-free, these coefficients are symmetric in $j,k$; polarization gives ${\Gamma }^i{}_{jk}(p)=0$. Metric compatibility then gives ${\partial }_k{g}_{ij}(p)=0$. These statements concern the center $p$; they do not make every translated coordinate line a geodesic.

The precise Gauss lemma isfor $v$ in such a sufficiently small neighborhood and every $w\in T_{p}M$. Thus the exponential map preserves the radial pairing; in particular it takes the radial direction orthogonally to the tangent directions of spheres about zero.

To prove it, form the smooth geodesic variationAll the curves with $s$ fixed are geodesics, so $D_{t}T=0$. The torsion-free Levi-Civita connection gives $D_{t}S=D_{s}T$ for this two-parameter map. This equality follows also by writing both derivatives in coordinates: the mixed partial derivatives agree and the Christoffel coefficients are symmetric. Metric compatibility yieldsThe constant-speed property permitted in the question gives $g(T,T)=g_p(v+sw,v+sw)$ for every $t$. Thus at $s=0$ the last expression is $g_p(v,w)$. Since $F(0,s)=p$, we have $S(0,s)=0$, and integrating in $t$ gives $g(T,S)(t,0)=t{g}_p(v,w)$. At $t=1$, the two variation derivatives are $T(1,0)=(d{\exp }_p{)}_{v}v$ and $S(1,0)=(d{\exp }_p{)}_{v}w$, proving the lemma. Its radial-length assertion follows by taking $w=v$, and its orthogonality assertion by taking $w\perp v$. Only constant speed is used here; the PDF correctly states constancy of the length of the velocity vector, whose dot is lost in the TeX aid.

A connection on a vector bundle specifies how to compare nearby fibers without identifying them globally. Several equivalent descriptions expose its algebra, local calculations and geometry.

The first description is a vector bundle covariant derivative, a real-linear mapEquivalently ${\nabla }_{X}s$ is linear over smooth functions in the vector field $X$, and obeys ${\nabla }_X(fs)=X(f)s+f{\nabla }_{X}s$ in the section. The derivative term distinguishes a connection from a bundle homomorphism. The Leibniz rule also makes it local: use a smooth cutoff equal to one near a point to see that its value depends only on the section near that point. Thus a connection restricts consistently to all open subsets.

The second description is a family of connection matrices in local frames. Write a frame as a row $e=(e_1,\dots ,e_m)$, so a section is $s=eu$ with coefficient column $u$. The connection is thenwhere $A$ is a matrix of one-forms determined by $\nabla e_j={\sum }_i{e}_i{A}^i{}_j$. If $e^{\prime }=eg$, the same section has $u=g{u}^{\prime }$. Expanding the Leibniz rule givesHence the change of frame of a vector-bundle connection isConversely, matrices of one-forms satisfying this transformation rule define $\nabla (eu)=e(du+Au)$ in each frame. The displayed calculation proves that these local formulas agree on overlaps; they therefore glue to a connection and obey its Leibniz rule. This proves equivalence of the first two descriptions in both directions.

A third description is parallel transport. Along a smooth curve $\gamma :[a,b]\to B$, a parallel section $s(t)=e(\gamma (t))u(t)$ satisfiesExistence and uniqueness for linear ordinary differential equations supply a unique solution for each initial fiber vector, smoothly dependent on that vector and on smooth families of curves. The frame transformation law makes these local solutions glue. The resulting map $P_{\gamma ,a,b}:E_{\gamma (a)}\to E_{\gamma (b)}$ is linear and invertible; reversing the curve gives its inverse, and concatenating curves composes the maps. It is generally dependent on the path, not just its endpoints.

One can recover the covariant derivative from this transport. For $\gamma (0)=b$ and $\dot{\gamma }(0)=X$, let $P_{0,t}$ be its transport from $0$ to $t$. ThenTo verify the formula rather than assume it, use a local frame and a fundamental solution matrix $P(t)$ with $P(0)=I$ and $\dot{P}=-A(\dot{\gamma })P$. Differentiation gives $(P^{-1}u{)}^{\prime }{|}_0=\dot{u}(0)+A_b(X)u(0)$, exactly the local covariant-derivative formula. Conversely the connection reconstructed from transport defined by this equation has the same differential equation and, by uniqueness, the same transport. Thus this viewpoint is also equivalent for the transport rules arising from a connection.

A fourth description is a linear horizontal distribution on a vector bundle. In a trivialization $E{|}_U=U\times {\mathbb{R}}^m$, putProjection maps this subspace isomorphically to $T_{b}B$, and it complements the vertical tangent space. Under the coordinate change $v=g{v}^{\prime }$, differentiation gives $\dot{v}=(dg)(X)v^{\prime }+g{\dot{v}}^{\prime }$. Thus the horizontal equation transforms into ${\dot{v}}^{\prime }=-A^{\prime }(X)v^{\prime }$, with exactly the same frame-change law. The subspaces consequently glue independently of trivializations. Their fiberwise linear dependence on $v$ is essential: an arbitrary horizontal complement can describe a nonlinear connection on the total space and need not define a vector-bundle connection. Conversely a horizontal complement with this linearity has a unique local expression of the displayed form, recovering the matrices $A$. Its horizontal lifts are precisely the parallel sections just described.

One can also express the geometry on the frame bundle as a principal connection. In local frame-bundle coordinates $(b,g)$, its Lie-algebra-valued form isIt is equivariant under right translation, $R_h^{\ast }\omega ={\mathrm{Ad}}_{h^{-1}}\omega$, and equals $\xi$ on the vertical vector generated by $\xi$. These properties define a principal connection; its kernel is the horizontal distribution on the frame bundle. Pulling back by a local frame section gives $A$, and its frame-change rule is once again the same formula. This identifies the principal viewpoint with the earlier descriptions.

Connections are not a vector space with a distinguished origin. The difference of two connections is linear over smooth functions in both $X$ and $s$, because their Leibniz terms cancel. It is a one-form with values in $\mathrm{End}E$. Conversely adding any such form to a connection gives another connection. Thus the collection is an affine space of vector-bundle connections modeled on ${\Omega }^1(B;\mathrm{End}E)$. The local curvature in the usual vector-bundle convention is $F=dA+A\wedge A$, representing ${\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$. It records the infinitesimal failure of path-independent parallel transport. Question 6 uses the opposite curvature sign, which will be stated explicitly there.