fields on differentiable manifolds · field theory

spacetime as a differentiable manifold

A differentiable manifold $M$ of dimension $n$ is a second-countable, Hausdorff topological space that is locally Euclidean: every point $p\in M$ possesses an open neighborhood $U$ together with a homeomorphism $\varphi:U\to\varphi(U)\subset\mathbb{R}^n$ called a chart. The collection of all such charts constitutes an atlas. Whenever two charts $(U,\varphi)$ and $(V,\psi)$ have nonempty intersection, the transition map $$ \psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V) $$ is required to be smooth (of class $C^\infty$). This smoothness condition guarantees that notions such as differentiability of functions, vector fields, and differential forms are intrinsically defined, independent of any particular choice of coordinates.

definition (smooth manifold). a second-countable Hausdorff space $M$ equipped with an atlas of charts $\varphi:U\to\mathbb{R}^n$ whose transition maps are $C^\infty$. two atlases define the same smooth structure if their union is again an atlas (they are equivalent under refinement).

definition (smooth map). a continuous map $f:M\to N$ between smooth manifolds is smooth if, for every pair of charts in which the composition is defined, the resulting map of open subsets of Euclidean spaces is $C^\infty$.

In the context of spacetime, $M$ is typically taken to be four-dimensional and orientable, though the geometric framework developed here applies equally to manifolds of arbitrary dimension. The rigid linear structure of Minkowski space is recovered only locally: each tangent space $T_p M$ is a vector space isomorphic to $\mathbb{R}^{1,3}$ (or $\mathbb{R}^n$ in the Riemannian setting), but the global topology and curvature of $M$ remain free. This liberation from a fixed background is essential for general relativity and for any field theory whose dynamics can deform the geometry itself.

definition (tangent space). the tangent space $T_p M$ is the vector space of derivations of the algebra of germs of smooth real functions at $p$ (equivalently, velocity vectors of smooth curves through $p$). the tangent bundle is $TM=\bigcup_{p\in M}T_p M$ with its natural smooth structure; its dual is the cotangent bundle $T^*M$.

remark (local minkowski structure). a Lorentzian metric (chapter 2) identifies each $T_p M$ with Minkowski space up to isometric isomorphism, but no preferred global isomorphism $M\simeq\mathbb{R}^{1,3}$ is assumed. field theory is written so that coordinate charts are tools, not ontology.

tensor algebras and the kinematics of classical fields

Once the manifold is given, the kinematic objects of classical field theory are constructed from the tangent and cotangent spaces. At each point $p$ one has $T_p M$ and its dual $T_p^* M$. The full tensor algebra is obtained by forming all finite tensor products of these spaces and then taking the direct sum over all ranks. Within this algebra three subalgebras are of special importance:

the contravariant tensor algebra generated by repeated tensor products of $T_p M$; the symmetric algebra, consisting of totally symmetric tensors (relevant for metrics and higher-rank gravitational potentials); and the exterior algebra $\bigwedge^\bullet T_p^*M$, consisting of totally antisymmetric tensors, the differential forms.

definition (tensor bundle). the bundle of tensors of type $(r,s)$ is $$ T^{r,s}M :=(TM)^{\otimes r}\otimes(T^*M)^{\otimes s}. $$ a classical tensor field of type $(r,s)$ is a smooth section of $T^{r,s}M$.

definition (section of a vector bundle). if $\pi:E\to M$ is a smooth vector bundle, a (smooth) section is a smooth map $s:M\to E$ with $\pi\circ s=\mathrm{id}_M$. the space of sections is written $\Gamma(E)$.

A classical field is, in the first instance, a smooth section of an associated vector bundle built from these constructions. Scalar fields are sections of the trivial line bundle; vector fields are sections of $TM$; differential forms of degree $k$ are sections of $\bigwedge^k T^*M$; and more elaborate fields (spinors, gauge potentials, and so on) arise as sections of bundles associated to principal bundles once the appropriate structure group is introduced. The tensorial character of these objects ensures that their transformation laws under changes of coordinates are completely determined by the transition functions of the manifold, rendering the description coordinate-free.

theorem (transformation law of a covector). let $(U,x^i)$ and $(V,y^a)$ be charts, and let $\alpha$ be a smooth one-form. writing $\alpha=\alpha_i\,dx^i=\tilde\alpha_a\,dy^a$ on $U\cap V$, $$ \tilde\alpha_a =\alpha_i\frac{\partial x^i}{\partial y^a}. $$ more generally, a type-$(r,s)$ tensor transforms with $r$ factors of $\partial y/\partial x$ and $s$ factors of $\partial x/\partial y$.

proof. by definition $dy^a=(\partial y^a/\partial x^i)\,dx^i$, so $$ \alpha =\alpha_i\,dx^i =\alpha_i\frac{\partial x^i}{\partial y^a}\,dy^a, $$ whence $\tilde\alpha_a=\alpha_i\,\partial x^i/\partial y^a$. the higher-rank case is the unique multilinear extension of this identity on pure tensors $X_1\otimes\cdots\otimes X_r\otimes \beta_1\otimes\cdots\otimes\beta_s$, extended by linearity and continuity. $\square$

remark (coordinate-free kinematics). once fields are sections of intrinsically defined bundles, equations written with abstract indices, exterior calculus, or principal-bundle connections automatically inherit the correct transformation laws. no separate covariance postulate is required beyond the geometry of $M$ and of the bundles over $M$.

exterior calculus of differential forms

Among all tensorial objects, differential forms occupy a privileged position because they admit a natural differential operator, the exterior derivative, that is independent of any metric or connection. If $\alpha$ is a $k$-form, its exterior derivative $d\alpha$ is the $(k+1)$-form defined locally by $$ d\alpha =\sum_I d\alpha_I\wedge dx^{i_1}\wedge\cdots\wedge dx^{i_k}, $$ where $\alpha_I$ are the component functions in a chart. The operator $d$ is nilpotent: $d^2=0$. This single algebraic property generates the entire de Rham complex $$ 0\to\Omega^0(M)\xrightarrow{d}\Omega^1(M)\xrightarrow{d}\cdots\xrightarrow{d}\Omega^n(M)\to 0. $$

definition (exterior derivative). $d:\Omega^k(M)\to\Omega^{k+1}(M)$ is the unique $\mathbb{R}$-linear antiderivation of degree one such that $d^2=0$ and $df$ agrees with the ordinary differential of a smooth function $f$.

theorem (nilpotence of $d$). for every differential form $\alpha$, $d(d\alpha)=0$.

proof. it is enough to check the claim in a chart, where any $k$-form is a sum of terms $f\,dx^{i_1}\wedge\cdots\wedge dx^{i_k}$. by the antiderivation rule and $d(dx^i)=0$, $$ d(f\,dx^{i_1}\wedge\cdots\wedge dx^{i_k}) =df\wedge dx^{i_1}\wedge\cdots\wedge dx^{i_k}, $$ and a second exterior derivative yields $$ d^2(f\,dx^I) =d(df)\wedge dx^I =\sum_{a,b}\partial_a\partial_b f\,dx^a\wedge dx^b\wedge dx^I. $$ the coefficient matrix $\partial_a\partial_b f$ is symmetric while $dx^a\wedge dx^b$ is skew, so the double sum vanishes. $\square$

The Poincare lemma asserts that every closed form ($d\alpha=0$) is locally exact: on any contractible open set there exists a form $\beta$ such that $\alpha=d\beta$. Globally, however, exactness may fail. The failure is measured by the de Rham cohomology groups $$ H^k_{\mathrm{dR}}(M) =\frac{\ker\bigl(d:\Omega^k\to\Omega^{k+1}\bigr)} {\operatorname{im}\bigl(d:\Omega^{k-1}\to\Omega^k\bigr)}. $$ These groups are topological invariants of $M$. They classify the obstructions to the existence of global potentials for given field strengths.

definition (closed and exact forms). a form $\alpha\in\Omega^k(M)$ is closed if $d\alpha=0$ and exact if $\alpha=d\beta$ for some $\beta\in\Omega^{k-1}(M)$. exact forms are closed by nilpotence of $d$; the converse is the content of local exactness and of vanishing of $H^k_{\mathrm{dR}}$.

theorem (poincare lemma). let $U\subset\mathbb{R}^n$ be open and star-shaped with respect to the origin (if $x\in U$ then $tx\in U$ for all $t\in[0,1]$). then $H^k_{\mathrm{dR}}(U)=0$ for every $k\ge 1$, and $H^0_{\mathrm{dR}}(U)=\mathbb{R}$. equivalently, every closed $k$-form on $U$ with $k\ge 1$ is exact.

proof. define a homotopy operator $K:\Omega^k(U)\to\Omega^{k-1}(U)$ for $k\ge 1$ by $$ (K\alpha)_x =\int_0^1 t^{k-1}\,\iota_X(\alpha_{tx})\,dt, $$ where $X=\sum_{i=1}^n x^i\partial_i$ is the radial vector field and $\iota_X$ denotes interior product. a direct computation with Cartan's formula along the straight-line homotopy $H(t,x)=tx$ yields the homotopy identity $$ dK+Kd=\mathrm{id}\qquad\text{on }\Omega^k(U),\ k\ge 1. $$ (write $\alpha$ in components, differentiate under the integral, and integrate by parts in $t$.) thus if $d\alpha=0$ then $\alpha=d(K\alpha)$. for $k=0$, closed zero-forms are locally constant functions; on a connected star-shaped set they are constant, so $H^0_{\mathrm{dR}}(U)=\mathbb{R}$. $\square$

In electromagnetism, for example, a closed two-form $F$ (the field strength) admits a global vector potential $A$ with $F=dA$ if and only if the de Rham class $[F]$ vanishes in $H^2_{\mathrm{dR}}(M)$. When the class is nontrivial one obtains magnetic monopoles or nontrivial flux sectors.

remark (local potentials versus global sectors). chart by chart the Poincare lemma always supplies a potential for a closed $F$. the obstruction is to glue those local potentials into a single global $A$, or equivalently into a connection on a $\mathrm{U}(1)$-bundle whose curvature is $F$. cohomology packages precisely that gluing obstruction.

computational tools: mayer-vietoris, stokes, and frobenius

The Mayer-Vietoris sequence provides a practical means of computing de Rham cohomology by decomposing $M$ into overlapping open sets whose individual cohomologies are known. If $M=U\cup V$, the sequence $$ \cdots\to H^k_{\mathrm{dR}}(M) \to H^k_{\mathrm{dR}}(U)\oplus H^k_{\mathrm{dR}}(V) \to H^k_{\mathrm{dR}}(U\cap V) \to H^{k+1}_{\mathrm{dR}}(M)\to\cdots $$ is exact and relates the global topology to local data.

theorem (mayer-vietoris for de rham cohomology). let $U,V\subset M$ be open with $M=U\cup V$. the sequence of de Rham groups displayed above is exact, where the maps are restriction $[\alpha]\mapsto([\alpha|_U],[\alpha|_V])$, difference $([\beta],[\gamma])\mapsto[\beta|_{U\cap V}-\gamma|_{U\cap V}]$, and the connecting homomorphism sending a closed form $\eta$ on $U\cap V$ to the class of a form obtained by gluing cutoffs of local primitives as follows.

proof. exactness at $H^k(U)\oplus H^k(V)$ is immediate: a pair of closed forms on $U$ and $V$ that agree on $U\cap V$ glue to a global closed form. for the connecting map, take a closed $\eta$ on $U\cap V$. let $\{\varphi_U,\varphi_V\}$ be a smooth partition of unity subordinate to $\{U,V\}$. set $\alpha_U:=-\varphi_V\eta$ on $U\cap V$ (extended by zero where $\varphi_V=0$ in $U$) and $\alpha_V:=\varphi_U\eta$ on $U\cap V$ (extended analogously in $V$). then on $U\cap V$, $$ \alpha_V-\alpha_U=(\varphi_U+\varphi_V)\eta=\eta, $$ so $d\alpha_U=d\alpha_V$ on the intersection (because $d\eta=0$). the common value defines a closed $(k+1)$-form $\omega$ on $M$, and $[\omega]\in H^{k+1}_{\mathrm{dR}}(M)$ is independent of the cutoffs. the remaining exactness statements are standard diagram chases with these explicit maps. $\square$

Integration of differential forms is governed by the generalized Stokes theorem: for an oriented $k$-dimensional manifold with boundary $\Sigma$, and $\alpha\in\Omega^{k-1}(\Sigma)$, $$ \int_\Sigma d\alpha=\int_{\partial\Sigma}\alpha $$ (with compact support, or suitable decay, understood). This identity is the geometric origin of all macroscopic conservation laws in classical field theory. The continuity equation, the integral form of Maxwell's equations, and the conservation of energy-momentum all appear as special cases once the appropriate forms are identified.

theorem (stokes). let $N$ be a compact oriented smooth $n$-manifold with boundary $\partial N$, and let $\alpha\in\Omega^{n-1}(N)$. then $$ \int_N d\alpha=\int_{\partial N}\alpha, $$ where $\partial N$ carries the induced orientation.

proof. cover $N$ by finitely many charts either diffeomorphic to open sets of $\mathbb{R}^n$ or to open sets of the half-space $\mathbb{H}^n=\{x^n\ge 0\}$. let $\{\psi_i\}$ be a partition of unity subordinate to the cover; then $$ \int_N d\alpha=\sum_i\int_N d(\psi_i\alpha),\qquad \int_{\partial N}\alpha=\sum_i\int_{\partial N}\psi_i\alpha, $$ so it is enough to prove the identity when $\alpha$ is supported in a single chart. in an interior chart the integral of $d\alpha$ is a sum of one-dimensional fundamental theorem of calculus contributions that cancel in pairs at opposite faces of a large cube containing the support. in a boundary chart the only unpaired face lies on $\{x^n=0\}$, and the corresponding FTCs produce exactly $\int_{\partial N}\alpha$ with the induced orientation. $\square$

Finally, the Frobenius theorem addresses the integrability of distributions. A subbundle $E\subset TM$ is integrable (that is, tangent to a foliation of $M$) if and only if it is closed under Lie bracket, or equivalently if the ideal of forms annihilating $E$ is differentially closed. In spacetime physics this criterion determines when a congruence of curves can be assembled into a family of hypersurfaces, a question that arises for trapped surfaces, event horizons, and the initial-value formulation of the field equations.

definition (distribution). a rank-$r$ distribution on $M$ is a smooth rank-$r$ subbundle $E\subset TM$. it is involutive if $[X,Y]\in\Gamma(E)$ whenever $X,Y\in\Gamma(E)$.

theorem (frobenius). a distribution $E\subset TM$ of constant rank is integrable, in the sense that every point has a neighborhood foliated by immersed submanifolds tangent to $E$, if and only if $E$ is involutive.

proof. necessity is elementary: if $E=T\mathcal{F}$ for a foliation $\mathcal{F}$, then Lie brackets of vector fields tangent to the leaves remain tangent to the leaves. for sufficiency, work locally and choose coordinates so that at $p$ one has $E_p=\operatorname{span} \{\partial_1,\dots,\partial_r\}$. involutivity and the straightening theorem for commuting frames (after correcting the frame by a lower-triangular change of generators that kills brackets, which is possible precisely because $[\partial_i,\partial_j]$ lies in $E$) produce a frame $Y_1,\dots,Y_r$ of $E$ with $[Y_i,Y_j]=0$. the joint flow of the commuting fields $Y_i$ defines a local immersion whose differential spans $E$, hence a local foliation. $\square$

local differential equations versus global topological constraints

The classical field equations themselves (Maxwell's equations, the Einstein equation, the Yang-Mills equation) are local differential equations on the manifold. Their solutions can be constructed chart by chart, subject only to the usual Cauchy or boundary-value problems. Yet the global topology of $M$ and of the bundles in which the fields take values imposes additional, nonlocal constraints. Cohomological obstructions may forbid the existence of global potentials; characteristic classes (to be introduced later) may quantize the total charge or the instanton number; and the fundamental group may force nontrivial holonomy around non-contractible loops.

Thus the complete description of a classical field configuration consists of two complementary layers: a local analytic layer governed by differential equations on open sets, and a global topological layer classified by cohomology classes, characteristic classes, and homotopy invariants. The subsequent chapters develop the geometric structures (connections, curvature, spinors, principal bundles) that make these two layers interact, culminating in the rich interplay between local dynamics and global topology that characterizes modern classical field theory.

remark (program of the chapter). this foundational chapter supplies the kinematic arena and the basic differential-topological language in which all later constructions are expressed. metrics and connections (chapter 2) will compare field values across paths; spinors (chapter 3) will lift the Lorentz structure; and characteristic classes with gauge dynamics will make the topological layer quantitative.

exercises

exercise 1 (transition maps). on the circle $S^1=\mathbb{R}/2\pi\mathbb{Z}$, write two angular charts that cover $S^1$ and compute the transition map on the overlap. verify that it is smooth as a map of open subsets of $\mathbb{R}$, and interpret the integer winding of that transition for the tangent bundle $TS^1$.

exercise 2 (poincare on a star domain). take $\alpha=x\,dy-y\,dx$ on $\mathbb{R}^2\setminus\{0\}$. show that $d\alpha=2\,dx\wedge dy$, so $\alpha$ is not closed. then for $\beta=(x\,dy-y\,dx)/(x^2+y^2)$ show $d\beta=0$ on the punctured plane, and argue that $\beta$ is not exact by integrating along the unit circle and applying Stokes on a disk.

exercise 3 (frobenius in coordinates). for the distribution on $\mathbb{R}^3$ spanned by $X=\partial_x+y\partial_z$ and $Y=\partial_y$, compute $[X,Y]$ and decide whether the distribution is integrable. if it is, exhibit local leaves; if not, exhibit the bracket obstruction explicitly.

exercise 4 (tensor transformation law). if $V=V^\mu\partial_\mu$ is a vector field and $x\mapsto y(x)$ is a coordinate change, derive $V^{\prime\nu}=\frac{\partial y^\nu}{\partial x^\mu}V^\mu$. state the dual law for a covector $\alpha=\alpha_\mu dx^\mu$ and check that $\alpha(V)$ is invariant.

exercise 5 (wedge product and interior product). on $\mathbb{R}^3$, let $\alpha=x\,dy$ and $\beta=dz$. compute $\alpha\wedge\beta$ and $i_{\partial_x}(\alpha\wedge\beta)$. verify the graded Leibniz rule $i_X(\alpha\wedge\beta)=(i_X\alpha)\wedge\beta+(-1)^{\deg\alpha}\alpha\wedge(i_X\beta)$.

exercise 6 (de rham versus singular (sketch)). using Stokes theorem, prove that if $\omega$ is exact then $\int_c\omega=0$ for every closed oriented cycle $c$. conclude that a closed form with a nonzero period cannot be exact (you may take the angle form of exercise 2 as the model case).