A velocity belongs to the tangent space at one point. A nearby velocity belongs to another tangent space. A connection specifies how to differentiate a vector field despite this change of space.
6.1. Differentiating vector fields
Write \(\mathfrak X(M)\) for smooth vector fields. A vector field acts on a smooth function by \((Xf)(p)=X_p(f)\). In coordinates, \(Xf=\sum_iX^i\partial_i f\), so this is smooth.
Definition 6.1. Lie bracket
The Lie bracket of vector fields is specified by \([X,Y]f=X(Yf)-Y(Xf)\).
Proposition 6.2. Bracket formula and product rules
The bracket is a smooth vector field with
\[[X,Y]^k=\sum_i(X^i\partial_iY^k-Y^i\partial_iX^k).\]It obeys \([X,Y]=-[Y,X]\), \([fX,Y]=f[X,Y]-Y(f)X\), and \([X,fY]=f[X,Y]+X(f)Y\). Coordinate fields commute: \([\partial_i,\partial_j]=0\).
Proof.
Expand \(X(Yf)-Y(Xf)\) in coordinates. The second derivatives cancel because mixed partials commute, leaving exactly the displayed first-order operator. Its coefficients are smooth, so it is a smooth vector field. Skew symmetry follows by reversing the subtraction. Expanding either function multiple with the scalar product rule gives both displayed rules. Constant coordinate coefficients give zero bracket.
□Example. On the plane, \([\partial_x,x\partial_y]=\partial_y\). Thus varying basis directions can have nonzero bracket even when the underlying space is flat.
Definition 6.3. Affine connection
An affine connection is an operation \((X,Y)\mapsto\nabla_XY\) on smooth vector fields, real bilinear, such that for smooth scalar \(f\),
\[\nabla_{fX}Y=f\nabla_XY,\qquad \nabla_X(fY)=X(f)Y+f\nabla_XY.\]We use these same operations on open subsets, with agreement under restriction. This local formulation avoids any extension of fields beyond their domains. In coordinates the Christoffel symbols are the smooth coefficients defined by \(\nabla_{\partial_i}\partial_j=\sum_k\Gamma^k_{ij}\partial_k\).
Linearity in \(X\) makes the differentiation direction depend only on its value at a point. The product rule in \(Y\) permits derivatives of the field being differentiated.
Proposition 6.4. Coordinate formula
Every connection obeys
\[(\nabla_XY)^k=\sum_iX^i\partial_iY^k+\sum_{i,j}\Gamma^k_{ij}X^iY^j.\]In a new coordinate system \(y\), its symbols satisfy
\[\widetilde\Gamma^a_{bc}=\sum_{k,i,j}\frac{\partial y^a}{\partial x^k}\frac{\partial x^i}{\partial y^b}\frac{\partial x^j}{\partial y^c}\Gamma^k_{ij}+\sum_k\frac{\partial y^a}{\partial x^k}\frac{\partial^2x^k}{\partial y^b\partial y^c}.\]The second derivative term is why Christoffel symbols are not tensor components.
Proof.
Expand \(X=\sum_iX^i\partial_i\), \(Y=\sum_jY^j\partial_j\), and apply the two connection rules. For the transition, substitute \(\partial_{y^b}=\sum_i(\partial x^i/\partial y^b)\partial_{x^i}\) in both slots. The product rule differentiates the coefficient of the second slot, giving \(\partial^2x^k/\partial y^b\partial y^c\). Convert the resulting \(x\)-basis vector to the \(y\)-basis to obtain the formula.
□Example. On the line with the flat connection \(\Gamma^x_{xx}=0\), use \(x=e^y\) on \(x>0\). The new coefficient is \((dy/dx)(d^2x/dy^2)=1\). Nonzero symbols alone therefore do not establish curvature.
Definition 6.5. Torsion and metric compatibility
The torsion of a connection is \(T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]\). The connection is torsion-free if \(T=0\). For a metric \(g\), the connection is metric-compatible if
\[X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla_XZ)\]for every triple of vector fields. A torsion-free metric-compatible connection is the Levi-Civita connection.
Proposition 6.6. Coordinate tests
Torsion is linear over smooth functions in both slots. Its components are \(T^k_{ij}=\Gamma^k_{ij}-\Gamma^k_{ji}\). Metric compatibility is equivalent to
\[\partial_i g_{jk}=\sum_\ell\left(\Gamma^\ell_{ij}g_{\ell k}+\Gamma^\ell_{ik}g_{j\ell}\right).\]Proof.
For example,
\[T(fX,Y)=f\nabla_XY-Y(f)X-f\nabla_YX-f[X,Y]+Y(f)X=fT(X,Y).\]Skew symmetry gives function-linearity in the other slot. Insert commuting coordinate fields to get the torsion components. For compatibility, insert three coordinate fields to obtain the displayed equation. Conversely expand arbitrary \(X,Y,Z\); derivatives of the components of \(Y,Z\) appear on both sides by the product rule and cancel, leaving that coordinate equation multiplied by \(X^iY^jZ^k\). Thus it suffices.
□Proposition 6.7. Existence and uniqueness of the Levi-Civita connection
Every smooth pseudo-Riemannian metric has a unique Levi-Civita connection, with coefficients
\[\Gamma^k_{ij}=\frac12\sum_\ell g^{k\ell}\left(\partial_i g_{j\ell}+\partial_j g_{i\ell}-\partial_\ell g_{ij}\right).\]Proof: the Koszul formula and gluing
Proof.
If a connection is compatible, write the compatibility identity for \((X,Y,Z)\), \((Y,X,Z)\), and \((Z,X,Y)\). Add the first two and subtract the third. Substitute \(\nabla_YX=\nabla_XY-[X,Y]\), and the corresponding torsion-free identities for the other pairs. Symmetry of \(g\) cancels the unwanted derivatives. The result is the Koszul formula:
\[\begin{aligned}2g(\nabla_XY,Z)={}&Xg(Y,Z)+Yg(X,Z)-Zg(X,Y)\\&-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]).\end{aligned}\]This proves uniqueness because nondegeneracy determines a vector from all its pairings with \(Z\). In coordinate fields the brackets vanish, and multiplication by the inverse metric gives exactly the displayed Christoffel formula.
For existence, define these smooth coefficients separately in each chart and use Proposition 6.4’s first formula to define a local connection. The symmetry in \(i,j\) makes it torsion-free. Multiplying by \(g_{k\ell}\) gives \(g(\nabla_{\partial_i}\partial_j,\partial_\ell)=\tfrac12(\partial_i g_{j\ell}+\partial_j g_{i\ell}-\partial_\ell g_{ij})\). Adding this expression to its version with \(j,\ell\) exchanged leaves \(\partial_i g_{j\ell}\); hence the compatibility test holds. On overlaps the two locally constructed operations are both compatible and torsion-free. The coordinate-independent Koszul formula just derived forces them to agree. Therefore they glue to a global connection with the required laws.
□Example. For constant metric coefficients, every derivative in the formula is zero, so \(\Gamma=0\). For the Euclidean plane in polar coordinates, \(g_{rr}=1\), \(g_{\theta\theta}=r^2\), and the other components vanish. The only nonzero symbols are \(\Gamma^r_{\theta\theta}=-r\) and \(\Gamma^\theta_{r\theta}=\Gamma^\theta_{\theta r}=1/r\), obtained by differentiating \(r^2\).
Example. Compatibility and torsion freedom are independent. On \(\mathbb R\) with \(g=dx^2\), the connection \(\nabla_XY=(XY'+XY)\partial_x\), writing \(X,Y\) for their scalar components, has \(\Gamma^x_{xx}=1\) and hence zero torsion. But compatibility would require \(0=2\), so it fails. Conversely, on Euclidean \(\mathbb R^2\), put \(\nabla_XY=D_XY+dx(X)JY\), where \(J(a,b)=(-b,a)\). The identity \(\langle JY,Z\rangle+\langle Y,JZ\rangle=0\) proves compatibility; its torsion has \(T(\partial_x,\partial_y)=-\partial_x\ne0\).
6.2. Derivatives along a curve
Definition 6.8. Covariant acceleration and geodesic
For a \(C^1\) vector field \(W(t)=\sum_kW^k(t)\partial_k|_{\gamma(t)}\) along a \(C^1\) curve, define
\[\left(\frac{DW}{dt}\right)^k=\frac{dW^k}{dt}+\sum_{i,j}\Gamma^k_{ij}(\gamma(t))\dot\gamma^iW^j.\]\(W\) is parallel if this derivative is zero. For a \(C^2\) curve its covariant acceleration is \(D\dot\gamma/dt\). A geodesic with affine parameter has zero covariant acceleration:
\[\ddot\gamma^k+\sum_{i,j}\Gamma^k_{ij}\dot\gamma^i\dot\gamma^j=0.\]Worked example for Definition 6.8
Counterexample: A straight path with a nonaffine parameter need not satisfy the geodesic equation
Proof.
Under a coordinate change, \(W_y^a=\sum_k(\partial y^a/\partial x^k)W_x^k\). Differentiate this identity along the curve. The derivative of its Jacobian supplies a second-coordinate-derivative term. Substituting the transformation law in Proposition 6.4 cancels that term, leaving \((DW/dt)_y^a=\sum_k(\partial y^a/\partial x^k)(DW/dt)_x^k\). Thus this definition is intrinsic; compare covariant differentiation along curves. For a compatible connection the same coordinate expansion as in Proposition 6.6 proves \(\tfrac d{dt}g(W,Z)=g(DW/dt,Z)+g(W,DZ/dt)\) along the curve.
□Example. With constant Minkowski metric, a geodesic has \(\ddot\gamma=0\), so it is \(\gamma(t)=p+tv\), by the mean value theorem applied twice to each component. For a proper-time-parametrized timelike curve, set \(A=DU/d\tau\). Differentiating \(g(U,U)=-c^2\) gives \(g(A,U)=0\). In flat affine coordinates this is the usual four-acceleration orthogonality.
6.3. Curvature
Definition 6.9. Curvature, Ricci tensor, and scalar curvature
Fix the sign convention
\[R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.\]The Ricci tensor is \(\operatorname{Ric}(Y,Z)=\operatorname{tr}(X\mapsto R(X,Y)Z)\). The scalar curvature is \(\operatorname{Scal}=\operatorname{tr}_g\operatorname{Ric}\). Trace of a linear endomorphism is the sum of its diagonal entries in a basis.
Proposition 6.10. Curvature is a tensor
\(R(X,Y)Z\) is linear over smooth functions in all three arguments and skew in \(X,Y\). Thus \((\alpha,X,Y,Z)\mapsto\alpha(R(X,Y)Z)\) is a smooth \((1,3)\)-tensor. Put \(R(\partial_i,\partial_j)\partial_k=\sum_\ell R^\ell{}_{kij}\partial_\ell\). Then
\[R^\ell{}_{kij}=\partial_i\Gamma^\ell_{jk}-\partial_j\Gamma^\ell_{ik}+\sum_a(\Gamma^a_{jk}\Gamma^\ell_{ia}-\Gamma^a_{ik}\Gamma^\ell_{ja}),\qquad \operatorname{Ric}_{jk}=\sum_iR^i{}_{kij}.\]Proof.
Skew symmetry follows from the definition. In \(R(fX,Y)Z\), the first term is \(f\nabla_X\nabla_YZ\); the second is \(-Y(f)\nabla_XZ-f\nabla_Y\nabla_XZ\). Since \([fX,Y]=f[X,Y]-Y(f)X\), the last term contributes \(-f\nabla_{[X,Y]}Z+Y(f)\nabla_XZ\). The extra terms cancel. Skew symmetry gives function-linearity in the second argument.
Expanding the third argument gives
\[R(X,Y)(fZ)=fR(X,Y)Z+\big(X(Yf)-Y(Xf)-[X,Y]f\big)Z=fR(X,Y)Z;\]the terms involving \(\nabla_XZ\) and \(\nabla_YZ\) cancel in pairs. Therefore the value at a point depends only on the three vectors there: express each field in a local coordinate basis and pull all its coefficients out. The coordinate formula follows by applying \(\nabla_{\partial_i}\) to \(\sum_a\Gamma^a_{jk}\partial_a\) and subtracting the reversed term. It has smooth coefficients. Taking the indicated matrix trace gives the Ricci formula. Trace is basis independent since \(\operatorname{tr}(AB)=\sum_{i,j}A_{ij}B_{ji}=\operatorname{tr}(BA)\), so conjugation preserves it. The metric trace is invariant by Proposition 5.6.
□Example. Constant \(\Gamma=0\) gives \(R=0\). In the polar-plane example,
\[R^r{}_{\theta r\theta}=\partial_r(-r)-\partial_\theta(0)+0-(-r)(1/r)=0.\]Every other component also vanishes because curvature is a tensor and it vanishes in Cartesian coordinates. Curvature detects a property of the connection, while Christoffel symbols also record the coordinates. We have defined geodesics and curvature; no existence theorem for geodesics or Einstein field equation is being assumed.
Worked examples
Each example checks the hypotheses and the calculation for the linked statements. The links in the statements return here.
Worked example 1. A nonzero bracket on the plane
Applies to: Definition 6.1, Proposition 6.2.
Let \(X=x\partial_y,Y=\partial_x\) on \(\mathbb R^2\). For every smooth \(f\), \(X(Yf)-Y(Xf)=xf_{xy}-(f_y+xf_{yx})=-f_y\). Therefore \([X,Y]=-\partial_y\), exactly the coordinate formula. Coordinate fields commute by equality of mixed partials. Multiplying \(X\) by the scalar function \(x\), the direct calculation gives \([x^2\partial_y,\partial_x]=-2x\partial_y\); the product rule gives the same \(x[X,Y]-Y(x)X=-2x\partial_y\). Skew symmetry gives the reversed bracket, and \([\partial_x,x^2\partial_y]=2x\partial_y\) checks the rule for multiplication in the second slot.
Worked example 2. The flat connection in polar coordinates
Applies to: Definition 6.3, Proposition 6.4, Definition 6.5, Proposition 6.6, Proposition 6.7.
On a polar chart \(r>0\) with an angular interval shorter than \(2\pi\), take \(g=dr^2+r^2d\theta^2\), the pullback of the Euclidean plane metric. Differentiating its components in the Levi-Civita formula gives the only nonzero coefficients \(\Gamma^r_{\theta\theta}=-r\), \(\Gamma^\theta_{r\theta}=\Gamma^\theta_{\theta r}=1/r\). Hence \(\nabla_{\partial_\theta}\partial_\theta=-r\partial_r\), \(\nabla_{\partial_r}\partial_\theta=r^{-1}\partial_\theta\). The coordinate formula is linear over functions in its first input and obeys the Leibniz rule in its second because the derivative term does, so these coefficients specify a connection.
The lower indices are symmetric, giving zero torsion. For compatibility the nonconstant component satisfies \(\partial_rg_{\theta\theta}=2r=2\Gamma^\theta_{r\theta}r^2\); the mixed test is \(\partial_\theta g_{r\theta}=0=\Gamma^\theta_{\theta r}r^2+\Gamma^r_{\theta\theta}=r-r\). All other tests are zero, so it is the unique Levi-Civita connection. Cartesian symbols are zero, but transforming them to polar coordinates includes \(\partial^2_\theta(r\cos\theta,r\sin\theta)=-(r\cos\theta,r\sin\theta)\), whose radial component is \(-r\). This checks the inhomogeneous transformation term and why these symbols are not tensor components.
Further proof details
Only \(\partial_rg_{\theta\theta}=2r\) is a nonzero metric-component derivative, and the inverse metric is \(\operatorname{diag}(1,r^{-2})\). Thus the coefficient formula gives \(\Gamma^r_{\theta\theta}=\tfrac12(0+0-2r)=-r\), \(\Gamma^\theta_{r\theta}=\tfrac12r^{-2}(2r+0-0)=1/r\), and its lower-index swap. Every remaining coefficient uses only zero derivatives. For the transformation check, \(dr=(\cos\theta,\sin\theta)\) as a Cartesian covector, so pairing it with \(\partial_{\theta}^2(x,y)=(-r\cos\theta,-r\sin\theta)\) gives \(-r\); pairing with \(d\theta=(-\sin\theta/r,\cos\theta/r)\) gives zero. These are the transformed coefficients for two angular inputs, obtained directly from the inhomogeneous term.
Worked example 3. A geodesic that is not a radial line
Applies to: Definition 6.8.
In this polar chart consider the Cartesian straight line \((x,y)=(t,1)\), on an interval near zero. Its polar coordinates have \(r=\sqrt{1+t^2}\), \(\theta=\pi/2-\arctan t\). Then \(r'=t/r,r''=1/r^3,\theta'=-1/r^2,\theta''=2t/r^4\). Substitution gives \(r''-r(\theta')^2=0\) and \(\theta''+2r'\theta'/r=0\). Thus its covariant acceleration vanishes with the affine parameter \(t\), although its coordinate second derivatives do not. Its tangent is a parallel vector field along the curve by precisely the same covariant-derivative equation, illustrating both parallel transport and the geodesic definition.
Worked example 4. A curved metric and its contractions
Applies to: Definition 6.9, Proposition 6.10.
On the unit sphere away from its poles, use \(0<\theta<\pi\) and a local longitude \(\phi\), with \(g=d\theta^2+\sin^2\theta\,d\phi^2\). The Levi-Civita coefficients are \(\Gamma^\theta_{\phi\phi}=-\sin\theta\cos\theta\) and \(\Gamma^\phi_{\theta\phi}=\Gamma^\phi_{\phi\theta}=\cot\theta\). With the chapter’s curvature convention,
\[R^\theta{}_{\phi\theta\phi}=\partial_\theta(-\sin\theta\cos\theta)-(-\sin\theta\cos\theta)\cot\theta=\sin^2\theta.\]Also \(R^\phi{}_{\theta\phi\theta}=-\partial_\theta\cot\theta-(\cot\theta)^2=1\). Hence \(\operatorname{Ric}_{\theta\theta}=1\), \(\operatorname{Ric}_{\phi\phi}=\sin^2\theta\), the mixed entries vanish, and \(\operatorname{Scal}=1+\sin^{-2}\theta\sin^2\theta=2\). Substitution of the coefficients into the other curvature components gives the compact formula \(R(X,Y)Z=g(Y,Z)X-g(X,Z)Y\). This expression is linear over smooth functions in all three fields and skew in the first two, explicitly displaying tensoriality and confirming the signs of the contractions.
Further proof details
The curvature check can be reduced to four explicit basis images. Put \(e=\partial_\theta,f=\partial_\phi\). From the displayed coefficients, \(\nabla_e e=0\), \(\nabla_e f=\nabla_f e=\cot\theta f\), and \(\nabla_f f=-\sin\theta\cos\theta e\). Since \([e,f]=0\),
\[R(e,f)e=\nabla_e(\cot\theta f)-\nabla_f0=(-\csc^2\theta+\cot^2\theta)f=-f,\] \[R(e,f)f=\nabla_e(-\sin\theta\cos\theta e)-\nabla_f(\cot\theta f)=\sin^2\theta e.\]Repeated first inputs give zero by the defining subtraction, and reversing them negates these values. Expanding the connection’s Leibniz rules cancels the derivatives of a multiplying scalar in each curvature slot, so these basis values determine all inputs. For \(Y=ae+bf,Z=ce+df\), \(R(e,Y)Z=b(d\sin^2\theta e-cf)\) and \(R(f,Y)Z=-a(d\sin^2\theta e-cf)\). Their diagonal coefficients as a map in the first argument sum to \(ac+bd\sin^2\theta=g(Y,Z)\). This proves \(\operatorname{Ric}=g\) and scalar curvature 2 with the declared sign, including the zero mixed entries.
Counterexamples and limits of the hypotheses
Each construction below refutes the explicitly stated stronger claim. It does not refute the related theorem with its original hypotheses.
Counterexample 1. A straight path with a nonaffine parameter need not satisfy the geodesic equation
Related statements: Definition 6.8.
Claim refuted. Every smooth parametrization of a geodesic image has zero covariant acceleration.
Proof.
Use the Euclidean plane’s flat Cartesian connection, so all Christoffel symbols are zero. The line \(\gamma(t)=(t,0)\) has zero acceleration. On \(s>0\), the increasing smooth reparametrization \(t=s^2\) has positive derivative, but \(\widetilde\gamma(s)=(s^2,0)\) has covariant acceleration \((2,0)\ne0\). It traces the same half-line with a different parameter. In general direct differentiation gives \(D(d\gamma/ds)/ds=(dt/ds)^2D\dot\gamma/dt+(d^2t/ds^2)\dot\gamma\). For a nonconstant affine geodesic only the first term vanishes automatically; an affine change of parameter also kills the second. Thus parameter independence of proper-time integrals does not imply parameter independence of the affine geodesic equation.
Counterexample 2. Zero curvature does not force zero torsion
Related statements: Definition 6.5, Definition 6.9.
Claim refuted. A flat connection must be torsion-free.
Proof.
On \(\mathbb R^2\) define \(\nabla_XY=D_XY+X^x JY\), where \(J\partial_x=\partial_y,J\partial_y=-\partial_x\), and \(D\) is ordinary Cartesian differentiation. Both terms are real bilinear; the additional term is linear over functions in both inputs, so the ordinary derivative’s two connection rules are preserved. The coefficient matrices are constant, with \(\nabla_{\partial_x}\) adding \(J\) and \(\nabla_{\partial_y}\) adding zero. For any smooth field \(Z\),
\[R(\partial_x,\partial_y)Z=(\partial_x+J)\partial_yZ-\partial_y(\partial_xZ+JZ)=0,\]since \(J\) is constant and mixed derivatives commute. Expanding the connection rules shows \(R(fX,Y)Z=fR(X,Y)Z\), and the same cancellation holds in the other slots; therefore this basis calculation gives \(R=0\). But \(T(\partial_x,\partial_y)=J\partial_y-0-0=-\partial_x\ne0\). In fact \(J\) is skew for the Euclidean dot product, so the added compatibility terms \(\langle JY,Z\rangle+\langle Y,JZ\rangle\) cancel. Even metric compatibility together with flatness does not supply torsion freedom.
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