Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 17 / 4

A connection on $TM$ is a metric connection for $g$ if for all smooth vector fields $X,Y,Z$,Equivalently, its induced covariant derivative of the metric satisfies $\nabla g=0$. This is also called a metric-compatible connection.

The torsion tensor of an affine connection isThe connection is symmetric, or a torsion-free connection, exactly whenIn a coordinate frame, write ${\nabla }_{{\partial }_i}{\partial }_j={\sum }_k{\Gamma }_{ij}^k{\partial }_k$. Since coordinate fields commute, the condition is ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$; this is the reason for the word symmetric.

The Levi-Civita connection is the unique affine connection which is both a metric connection and a torsion-free connection. Uniqueness is encoded by the Koszul formulaIt follows by adding the metric-compatibility identities with derivative fields $X,Y$, subtracting the one with derivative field $Z$, and replacing differences of covariant derivatives by Lie brackets. Nondegeneracy of $g$ determines ${\nabla }_{X}Y$ uniquely from all its pairings with $Z$. The existence and uniqueness of the Levi-Civita connection supplies existence; no existence proof is needed here.

In Cartesian coordinates on Euclidean space, defineThis is an affine connection by the ordinary product rule. Its connection coefficients in this coordinate frame are all zero. The component formula for the Lie bracket of vector fields gives ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$, so it is torsion-free. Also, since the Euclidean metric has constant coefficients,This is exactly metric compatibility. By uniqueness it is the Euclidean Levi-Civita connection. In particular constant vector fields are parallel, while the derivative of a general field is given by the boxed formula.

In coordinates, write ${\nabla }_{{\partial }_i}{\partial }_{\ell }={\Gamma }_{i\ell }^k{\partial }_k$. The Riemannian curvature two-form is defined by the matriceswith repeated indices summed. It assigns to $X,Y$ the fiberwise endomorphism $R(X,Y)$ whose entries are $R^k{}_{\ell ij}{X}^i{Y}^j$. We can prove both coordinate independence and the required intrinsic identity as follows.

Define an operator on vector fields by $C(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}$. It is alternating in $X,Y$. For a smooth function $f$, the identities $[fX,Y]=f[X,Y]-Y(f)X$ and ${\nabla }_Y(fZ)=Y(f)Z+f{\nabla }_{Y}Z$ givethe two terms involving $Y(f){\nabla }_{X}Z$ cancel. Alternation gives linearity over smooth functions in $Y$ as well. Expanding in the final argument yieldsThus $C$ is tensorial in all three arguments, defining a global endomorphism-valued two-form.

For coordinate fields ${\partial }_i,{\partial }_j$, their bracket is zero. Expand $[{\nabla }_{{\partial }_i},{\nabla }_{{\partial }_j}]Z$ using $Z=Z^{\ell }{\partial }_{\ell }$. The second derivatives of $Z^{\ell }$ cancel by equality of mixed partials. The terms containing first derivatives of $Z^{\ell }$ cancel in pairs, and the remaining coefficient of $Z^{\ell }{\partial }_k$ is exactly $R^k{}_{\ell ij}$ above. Hence the coordinate definition equals the global tensor $C$, establishing coordinate independence and provingThis convention for the Riemann curvature tensor is valid for any affine connection; in particular it applies to the Levi-Civita connection.

Take $\mathbf{n}$ to be the unit normal, so $P(v)=v-⟨v,\mathbf{n}⟩\mathbf{n}$ is orthogonal projection onto $TM$. With $D$ the Euclidean Levi-Civita connection, the proposed derivative isIt is a smooth tangent field, and independence of the extensions is given. Linearity follows from linearity of $D$ and $P$. Moreover, for $f\in C^{\infty }(M)$, take local extensions of $f,X,Y$ to obtainsince $PY=Y$. Thus it is an affine connection on $M$.

For tangent fields $Y,Z$, the normal component of $DY$ pairs to zero with $Z$. Differentiating the ambient inner product along $X$ givesHence it is a metric connection for the induced Riemannian metric. The ambient torsion vanishes, soHere the restriction of the ambient bracket is the intrinsic bracket and is tangent to $M$: derivations of functions restricted to $M$ give the same bracket, or this follows in submanifold coordinates. Thus the projected derivative is also a torsion-free connection. Uniqueness now provesThis is the projected ambient connection. The projection formula requires a unit normal. If a nonunit normal is used, its normal term must instead be $⟨DY,\mathbf{n}⟩\mathbf{n}/⟨\mathbf{n},\mathbf{n}⟩$. For example, on the unit circle, taking $\mathbf{n}=2p$ and differentiating its unit tangent along itself gives $DY=-p$; the unnormalized printed expression would be $3p$, which is not tangent. Normalizing the normal removes that ambiguity.