2 · metrics, connections, and curvature

pseudo-riemannian metrics and the levi-civita connection

A pseudo-Riemannian metric on an $n$-dimensional manifold $M$ is a smooth, symmetric, nondegenerate bilinear form $g$ of fixed signature on each tangent space. In spacetime the signature is Lorentzian, conventionally $(-,+,+,+)$ or $(+,-,-,-)$. The metric defines the causal structure, proper time along timelike curves, and the Hodge dual on differential forms.

definition (pseudo-riemannian metric). a smooth section $g$ of $\operatorname{Sym}^2 T^*M$ such that each $g_p:T_p M\times T_p M\to\mathbb{R}$ is a nondegenerate symmetric bilinear form of constant signature. if the signature is definite one speaks of a Riemannian metric; if it is Lorentzian one speaks of a spacetime metric.

definition (causal character). a nonzero vector $v\in T_p M$ is timelike, null, or spacelike according as $g_p(v,v)$ is negative, zero, or positive (in the signature $(-,+,+,+)$). a curve is classified by the character of its velocity.

Among all linear connections compatible with a given metric, there is a unique torsion-free connection: the Levi-Civita connection $\nabla$. Its Christoffel symbols in local coordinates are $$ \Gamma^\lambda_{\mu\nu} =\frac12 g^{\lambda\sigma}\bigl( \partial_\mu g_{\nu\sigma}+\partial_\nu g_{\mu\sigma}-\partial_\sigma g_{\mu\nu}\bigr). $$ Metric compatibility means $\nabla g=0$, so lengths and angles are preserved under parallel transport. Vanishing torsion means that the antisymmetric part of the connection coefficients is zero, equivalently that the torsion tensor $$ T(X,Y)=\nabla_X Y-\nabla_Y X-[X,Y] $$ vanishes identically.

definition (affine connection). an affine connection on $M$ is an $\mathbb{R}$-bilinear operator $\nabla:\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathfrak{X}(M)$, written $\nabla_X Y$, which is $C^\infty$-linear in $X$ and satisfies the Leibniz rule $\nabla_X(fY)=X(f)Y+f\nabla_X Y$.

definition (metric compatibility and torsion). $\nabla$ is metric-compatible if $Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)$ for all vector fields, written $\nabla g=0$. the torsion is $T(X,Y)=\nabla_X Y-\nabla_Y X-[X,Y]$.

theorem (fundamental theorem of riemannian geometry). on a pseudo-Riemannian manifold $(M,g)$ there exists a unique affine connection $\nabla$ that is torsion-free and metric-compatible. it is called the Levi-Civita connection of $g$.

proof. assume $\nabla$ is torsion-free and metric-compatible. expand the three cyclic permutations of metric compatibility, $$ \begin{aligned} Xg(Y,Z)&=g(\nabla_X Y,Z)+g(Y,\nabla_X Z),\\ Yg(Z,X)&=g(\nabla_Y Z,X)+g(Z,\nabla_Y X),\\ Zg(X,Y)&=g(\nabla_Z X,Y)+g(X,\nabla_Z Y). \end{aligned} $$ add the first two, subtract the third, and replace $\nabla_Y X=\nabla_X Y-[X,Y]$ (and cyclic analogs) by torsion-freeness. the result is Koszul's formula $$ \begin{aligned} 2g(\nabla_X Y,Z) &=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\\ &\quad -g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]). \end{aligned} $$ the right-hand side depends only on $g$ and the Lie bracket, so $g(\nabla_X Y,Z)$ is uniquely determined for all $Z$. nondegeneracy of $g$ therefore determines $\nabla_X Y$ uniquely. for existence, declare $\nabla_X Y$ by Koszul's formula in any chart (or abstractly by the same identity); the right-hand side is $C^\infty$-linear in $Z$, hence defines a vector field, and the usual verifications show that this $\nabla$ is an affine connection that is torsion-free and metric-compatible. in coordinates the Koszul prescription reduces to the displayed Christoffel formula. $\square$

Parallel transport along a curve $\gamma$ is the unique solution of the first-order equation $\nabla_{\dot\gamma}V=0$. Geodesics are the autoparallel curves satisfying $\nabla_{\dot\gamma}\dot\gamma=0$; they extremize the proper-time functional (for timelike curves in Lorentzian signature) and furnish the trajectories of freely falling test particles.

definition (parallel transport and geodesic). a vector field $V$ along a curve $\gamma$ is parallel if $\nabla_{\dot\gamma}V=0$. the curve $\gamma$ is a geodesic if $\nabla_{\dot\gamma}\dot\gamma=0$.

theorem (existence and uniqueness of parallel transport). let $\gamma:[a,b]\to M$ be a piecewise $C^1$ curve and let $v\in T_{\gamma(a)}M$. there exists a unique piecewise $C^1$ vector field $V$ along $\gamma$ with $V(a)=v$ and $\nabla_{\dot\gamma}V=0$ on each smooth piece.

proof. in a chart along a smooth piece the equation $\nabla_{\dot\gamma}V=0$ becomes the linear ODE $$ \dot V^\lambda+\Gamma^\lambda_{\mu\nu}(\gamma)\dot\gamma^\mu V^\nu=0. $$ existence and uniqueness for linear first-order ODEs with continuous coefficients yield a unique solution on that piece. continuous matching of the vector at breakpoints gives the unique solution on $[a,b]$. $\square$

remark (metric preservation). if $\nabla$ is the Levi-Civita connection and $V,W$ are parallel along $\gamma$, then $\frac{d}{dt}g(V,W)=0$. thus parallel transport is a linear isometry $T_{\gamma(a)}M\to T_{\gamma(b)}M$.

the riemann curvature tensor and the bianchi identities

Curvature measures the failure of parallel transport to be path-independent. The Riemann curvature tensor is defined by the commutator of covariant derivatives: $$ R(X,Y)Z=\nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z-\nabla_{[X,Y]}Z. $$ In components, $$ R^\rho{}_{\sigma\mu\nu} =\partial_\mu\Gamma^\rho_{\nu\sigma} -\partial_\nu\Gamma^\rho_{\mu\sigma} +\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}. $$

definition (riemann, ricci, scalar curvature). $R$ is the $(1,3)$-tensor determined by the formula above. the lowered tensor is $R_{\rho\sigma\mu\nu}=g_{\rho\lambda}R^\lambda{}_{\sigma\mu\nu}$. the Ricci tensor is $\operatorname{Ric}_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}$, and the scalar curvature is $\mathrm{Scal}=g^{\sigma\nu}\operatorname{Ric}_{\sigma\nu}$.

The tensor satisfies the algebraic symmetries $$ R_{\rho\sigma\mu\nu}=-R_{\sigma\rho\mu\nu}=-R_{\rho\sigma\nu\mu}=R_{\mu\nu\rho\sigma} $$ and the first Bianchi identity $$ R^\rho{}_{\sigma\mu\nu}+R^\rho{}_{\mu\nu\sigma}+R^\rho{}_{\nu\sigma\mu}=0. $$ The second (differential) Bianchi identity $$ \nabla_\lambda R^\rho{}_{\sigma\mu\nu} +\nabla_\mu R^\rho{}_{\sigma\nu\lambda} +\nabla_\nu R^\rho{}_{\sigma\lambda\mu}=0 $$ is a geometric constraint of profound physical consequence. When the Einstein equation is imposed, the contracted Bianchi identity implies the covariant conservation of the energy-momentum tensor, $$ \nabla_\mu T^{\mu\nu}=0, $$ thereby guaranteeing local energy-momentum balance without additional assumptions.

theorem (algebraic symmetries of the riemann tensor). for the Levi-Civita connection of a pseudo-Riemannian metric, $$ R_{\rho\sigma\mu\nu}=-R_{\sigma\rho\mu\nu}=-R_{\rho\sigma\nu\mu},\qquad R_{\rho\sigma\mu\nu}=R_{\mu\nu\rho\sigma}, $$ and the first Bianchi identity $R^\rho{}_{\sigma\mu\nu}+R^\rho{}_{\mu\nu\sigma}+R^\rho{}_{\nu\sigma\mu}=0$ holds.

proof. skew-symmetry in $(X,Y)$ is immediate from the definition of $R(X,Y)Z$. metric compatibility implies $$ Xg(\nabla_Y Z,W)+Xg(Z,\nabla_Y W)=XYg(Z,W) $$ and cyclic rearrangements; evaluating the cyclic sum that defines $g(R(X,Y)Z,W)+g(Z,R(X,Y)W)$ yields zero, which is $R_{\rho\sigma\mu\nu}=-R_{\sigma\rho\mu\nu}$ after lowering. the pair-swap symmetry $R_{\rho\sigma\mu\nu}=R_{\mu\nu\rho\sigma}$ follows from the first Bianchi identity together with the two skew-symmetries by a standard algebraic identity (expand the three cyclic sums with lowered indices). for the first Bianchi identity in the torsion-free case, expand $$ \mathfrak{S}_{X,Y,Z}R(X,Y)Z =\mathfrak{S}_{X,Y,Z}\bigl(\nabla_X\nabla_Y Z-\nabla_Y\nabla_X Z-\nabla_{[X,Y]}Z\bigr) $$ and cancel terms by the torsion-free Jacobi identity for the Lie bracket; the sum vanishes identically. $\square$

theorem (second bianchi identity). for the Levi-Civita connection, $$ \mathfrak{S}_{X,Y,Z}(\nabla_X R)(Y,Z)=0, $$ which in components is the differential Bianchi identity displayed above.

proof. work in a normal neighborhood of $p$ with a geodesic frame at $p$ so that $\Gamma^\lambda_{\mu\nu}(p)=0$ and therefore covariant derivatives of tensors at $p$ reduce to partial derivatives of components. the definition of $R$ then gives, at $p$, $$ (\nabla_X R)(Y,Z)W =X\bigl(R(Y,Z)W\bigr)-R(\nabla_X Y,Z)W-\cdots $$ with all $\nabla$-terms on frame fields vanishing at $p$. expanding $\mathfrak{S}_{X,Y,Z}\nabla_X\nabla_Y\nabla_Z W$ by associativity and canceling against the commutator definitions of $R$ produces zero at $p$. since $p$ was arbitrary, the identity holds everywhere. $\square$

theorem (contracted bianchi and conservation). the contracted second Bianchi identity reads $$ \nabla^\mu\bigl(\operatorname{Ric}_{\mu\nu}-\tfrac12\mathrm{Scal}\,g_{\mu\nu}\bigr)=0. $$ consequently, if Einstein's equation $\operatorname{Ric}_{\mu\nu}-\frac12\mathrm{Scal}\,g_{\mu\nu} +\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}$ holds, then $\nabla^\mu T_{\mu\nu}=0$.

proof. contract the second Bianchi identity on the first and third indices (relabel carefully) and use the algebraic symmetries to obtain $\nabla^\rho\operatorname{Ric}_{\rho\nu}=\frac12\nabla_\nu\mathrm{Scal}$. rearranging yields divergence-freeness of the Einstein tensor $G_{\mu\nu}=\operatorname{Ric}_{\mu\nu} -\frac12\mathrm{Scal}\,g_{\mu\nu}$. the cosmological term $\Lambda g_{\mu\nu}$ is parallel for the Levi-Civita connection, so $\nabla^\mu G_{\mu\nu}+\Lambda\nabla^\mu g_{\mu\nu}=0$ still holds. Einstein's equation therefore forces $\nabla^\mu T_{\mu\nu}=0$. $\square$

remark (holonomy infinitesimal). if $P_\gamma$ denotes parallel transport around a small parallelogram spanned by $X,Y$, then $P_\gamma=\mathrm{id}-R(X,Y)+o(|X||Y|)$. curvature is the infinitesimal holonomy of the connection.

ehresmann connections on principal fiber bundles

The notion of connection extends far beyond the tangent bundle. Let $\pi:P\to M$ be a principal $G$-bundle. An Ehresmann connection is a smooth horizontal distribution $H\subset TP$ complementary to the vertical distribution $V=\ker d\pi$ and invariant under the right $G$-action. Equivalently, a connection may be specified by a $\mathfrak{g}$-valued one-form $\omega$ on $P$ that reproduces the generators of the fundamental vector fields and transforms under the adjoint representation.

definition (principal $G$-bundle). a smooth free right $G$-action on a manifold $P$ with quotient map $\pi:P\to M=P/G$ a smooth submersion, locally trivial as $P|_U\simeq U\times G$ by $G$-equivariant diffeomorphisms.

definition (ehresmann connection / connection form). an Ehresmann connection is a smooth complementary horizontal distribution $H\subset TP$ with $T_u P=H_u\oplus V_u$ and $R_g^*H=H$ for all $g\in G$. equivalently, a connection one-form is $\omega\in\Omega^1(P,\mathfrak{g})$ such that $\omega(\xi_P)=\xi$ for every fundamental field $\xi_P$ generated by $\xi\in\mathfrak{g}$, and $R_g^*\omega=\operatorname{Ad}_{g^{-1}}\omega$. the horizontal space is $H_u=\ker\omega_u$.

definition (curvature form). the curvature of a connection form $\omega$ is the $\mathfrak{g}$-valued two-form $$ \Omega=d\omega+\tfrac12[\omega,\omega], $$ equivalently $\Omega(X,Y)=d\omega(X^H,Y^H)$ for the horizontal projections of $X,Y$.

Parallel displacement of a point $u\in P$ along a curve $\gamma$ in $M$ is obtained by lifting $\gamma$ to a horizontal curve in $P$. The holonomy group $\mathrm{Hol}_u(P,\omega)$ consists of all elements of $G$ realized by parallel transport around loops based at $\pi(u)$. The holonomy theorem of Ambrose and Singer identifies the Lie algebra of the holonomy group with the algebra generated by the curvature form evaluated on horizontal vectors. The reduction theorem asserts that if the holonomy group lies in a closed subgroup $H\subset G$, then the structure group of $P$ can be reduced to $H$.

theorem (existence of horizontal lifts). let $\omega$ be a connection on $P\to M$ and let $\gamma:[0,1]\to M$ be piecewise $C^1$ with $\gamma(0)=\pi(u)$. there exists a unique piecewise $C^1$ horizontal lift $\tilde\gamma$ of $\gamma$ with $\tilde\gamma(0)=u$, meaning $\pi\circ\tilde\gamma=\gamma$ and $\omega(\dot{\tilde\gamma})=0$.

proof. locally trivialize $P|_U\simeq U\times G$ so that $\omega$ becomes a $\mathfrak{g}$-valued one-form $A$ on $U$ (plus the Maurer-Cartan form on $G$). a horizontal lift is then a path $(\gamma(t),g(t))$ satisfying the linear ODE $\dot g=-A(\dot\gamma)g$ (matrix form) or the corresponding right-invariant equation on $G$. unique solutions for ODEs on $G$, together with unique continuation across a finite cover of $\gamma$, give the global horizontal lift. $\square$

theorem (ambrose-singer, statement). the Lie algebra of $\mathrm{Hol}_u^0(P,\omega)$ (the restricted holonomy group of piecewise smooth null-homotopic loops) is generated by all elements $\Omega_v(X,Y)$ where $v$ lies in the horizontal holonomy-leaf through $u$ and $X,Y\in H_v$.

proof (outline of idea). parallel transport around an infinitesimal parallelogram produces the curvature as in the Riemann case. the Lie algebra of a Lie group is recovered from one-parameter subgroups generated by such infinitesimal conjugations; integrating curvature along a dense set of loops fills the holonomy algebra. a complete rigorous treatment uses the reduction of the bundle to the holonomy group and the fact that curvature takes values in that algebra on the reduced total space. $\square$

theorem (holonomy reduction). if $H\subset G$ is a closed subgroup containing $\mathrm{Hol}_u(P,\omega)$, then the set of points of $P$ reachable from $u$ by horizontal curves is a principal $H$-subbundle to which $\omega$ reduces.

proof. let $Q$ be the set of endpoints of horizontal lifts starting at $u$. if $\tilde\gamma$ ends at $v$ and $h\in H$ contains the holonomy that would correct any two paths to the same basepoint, the $H$-orbit of $Q$ is well-defined and free. local triviality follows because horizontal lifts vary smoothly with basepaths in charts. by construction $\omega|_Q$ takes values in $\mathfrak{h}$ on vertical vectors for $H$, so the connection reduces. $\square$

remark (gauge potentials). pullback of $\omega$ by a local section $s:U\to P$ yields a local $\mathfrak{g}$-valued one-form $A=s^*\omega$ on $U$, the gauge potential. under $s'=s\cdot g$ one has $A'=g^{-1}Ag+g^{-1}dg$, recovering the classical gauge transformation law.

completeness, decomposition, and space forms

A Riemannian manifold is geodesically complete if every geodesic extends to all values of its affine parameter. The Hopf-Rinow theorem equates geodesic completeness with metric completeness (every Cauchy sequence converges) and guarantees that any two points may be joined by a minimizing geodesic. In the Lorentzian setting the corresponding notions are more subtle, but geodesic incompleteness remains the standard geometric signal of singularities.

definition (geodesic completeness). $(M,g)$ is geodesically complete if every maximal geodesic is defined on all of $\mathbb{R}$.

theorem (hopf-rinow). for a connected Riemannian manifold $(M,g)$ the following are equivalent: (i) $M$ is geodesically complete; (ii) $(M,d_g)$ is a complete metric space; (iii) closed bounded subsets of $M$ are compact. if any holds, then any two points of $M$ may be joined by a length-minimizing geodesic.

proof. the Riemannian distance $d_g(p,q)$ is the infimum of lengths of piecewise $C^1$ curves from $p$ to $q$. geodesic completeness implies metric completeness because a Cauchy sequence eventually lies in a compact geodesic ball once the exponential map is defined for long enough times (Hopf-Rinow's standard argument: extend geodesics from a fixed basepoint to reach limit points). conversely, if a geodesic cannot be extended past finite affine time, the sequence of points along it at rational times is Cauchy but has no limit in $M$. once metric completeness holds, the exponential map at $p$ is defined on all of $T_p M$; the usual length comparison shows that a point $q$ is reached by a minimizing geodesic realizing $d_g(p,q)$. compact closed balls follow from Hopf-Rinow's characterization via completeness of the metric topology. $\square$

The de Rham decomposition theorem states that a simply connected, complete Riemannian manifold whose holonomy representation is reducible splits isometrically as a product of lower-dimensional factors. Space forms (manifolds of constant sectional curvature) are locally isometric to Euclidean space, the sphere, or hyperbolic space according to the sign of the curvature; their global geometry is completely classified by the fundamental group.

theorem (de rham decomposition, statement). let $(M,g)$ be a simply connected, complete Riemannian manifold. if the holonomy representation of $T_p M$ decomposes as an orthogonal sum of irreducible subspaces, then $M$ is isometric to a Riemannian product of complete factors whose tangent spaces realize those summands.

proof (idea). parallel transport preserves the holonomy-invariant decomposition of $T_p M$, yielding smooth complementary parallel distributions $E$ and $F$ on $M$. by the Frobenius theorem (chapter 1) these integrate to foliations; completeness and simple connectivity imply that $M$ is a global product and the metric is a product metric. $\square$

Infinitesimal affine transformations are vector fields that preserve the connection; they generate the Lie algebra of the affine group of the manifold and play a central role in the study of homogeneous spaces and symmetric spaces.

definition (affine vector field). a vector field $X$ is affine for $\nabla$ if its flow preserves $\nabla$, equivalently $\mathcal{L}_X\nabla=0$ (in components, $\nabla_\mu\nabla_\nu X^\rho+R^\rho{}_{\sigma\mu\nu}X^\sigma=0$).

curvature as the obstruction to flatness

A connection is flat when its curvature vanishes identically. Locally, flatness is equivalent to the existence of parallel frames; globally, the holonomy representation must be trivial. Sectional curvature $K(\Pi)$ of a two-plane $\Pi\subset T_p M$ is the Gauss curvature of the surface obtained by exponentiating $\Pi$. When the sectional curvature is constant, the manifold is a space form; the comparison of local sectional curvature with global isometries yields rigidity theorems (for example, the Cartan-Hadamard theorem for nonpositive curvature and the sphere theorem for positive curvature).

definition (sectional curvature). for a nondegenerate plane $\Pi=\operatorname{span} \{X,Y\}\subset T_p M$, $$ K(\Pi) =\frac{g\bigl(R(X,Y)Y,X\bigr)}{g(X,X)g(Y,Y)-g(X,Y)^2}. $$

theorem (flatness and local parallel frames). let $\nabla$ be an affine connection on $M$. the following are equivalent in a simply connected open set $U$: (i) $R\equiv 0$ on $U$; (ii) every $p\in U$ admits a neighborhood with a parallel frame field; (iii) parallel transport in $U$ is path-independent.

proof. if a parallel frame $e_i$ exists, then $R(X,Y)e_i=0$ for all $X,Y$, so $R=0$. conversely, if $R=0$ then the PDE $\nabla e_i=0$ is integrable by the Frobenius theorem on the frame bundle (the horizontal distribution of the induced linear connection is involutive precisely when $R=0$). unique continuation of parallel transport from a chosen frame at $p$ then yields a local parallel frame. path-independence of parallel transport on loops contractible in $U$ follows, and on a simply connected $U$ this gives global path-independence within $U$. $\square$

theorem (cartan-hadamard, statement). if $(M,g)$ is a complete, simply connected Riemannian manifold of nonpositive sectional curvature, then $\exp_p:T_p M\to M$ is a diffeomorphism for every $p$, and $M$ is diffeomorphic to $\mathbb{R}^n$.

proof (outline). nonpositive curvature implies, by the second variation formula / Jacobi field comparison, that $\exp_p$ has no conjugate points. geodesic completeness and the Hopf-Rinow theorem make $\exp_p$ a smooth covering map; simple connectivity forces it to be a diffeomorphism. $\square$

Thus curvature simultaneously obstructs the existence of global parallel frames, dictates the focusing of geodesics, constrains the possible holonomy groups, and, through the Bianchi identities, enforces the conservation laws that underwrite classical field dynamics. The geometric structures introduced in this chapter (metric, Levi-Civita connection, curvature, and their principal-bundle generalizations) constitute the indispensable bridge between the local differential equations of classical fields and the global topology of spacetime.

remark (bridge to later chapters). spin geometry will lift the orthonormal frame bundle to a spin cover (chapter 3); gauge fields will treat connections on general principal bundles as dynamical variables (chapters 4 and 9); characteristic classes will be built from curvature forms by Chern-Weil theory (chapter 6).

exercises

exercise 1 (koszul). from Koszul's formula deduce the Christoffel formula for $\Gamma^\lambda_{\mu\nu}$ in a coordinate frame $\partial_\mu$, using $g(\nabla_{\partial_\mu}\partial_\nu,\partial_\sigma)=\Gamma^\lambda_{\mu\nu}g_{\lambda\sigma}$.

exercise 2 (geodesic conservation). if $X$ is a Killing field ($\mathcal{L}_X g=0$) and $\gamma$ is a geodesic, show that $g(\dot\gamma,X)$ is constant along $\gamma$. interpret as a Noether charge of the geodesic Lagrangian.

exercise 3 (u(1) holonomy). on $S^1$ with the flat connection $A=a\,d\theta$ ($a\in\mathbb{R}$) on the trivial $\mathrm{U}(1)$ bundle, compute the holonomy of the loop that winds once. for which $a$ is the holonomy trivial?

exercise 4 (curvature of a surface of revolution). for the metric $ds^2=dr^2+r^2 d\theta^2$ on the plane in polar coordinates, compute the only independent Christoffel symbols and show that the Gaussian curvature vanishes. explain why this is consistent with flatness of $\mathbb{R}^2$.

exercise 5 (parallel transport around a loop). on the unit sphere with the round metric, transport a unit tangent vector once around a latitude at polar angle $\vartheta$. show that the holonomy angle equals the solid angle of the enclosed spherical cap, $2\pi(1-\cos\vartheta)$.