Connections

Abstract

Before defining a notion of curvature that makes sense on arbitrary Riemannian manifolds, we need to study geodesics, the generalizations to Riemannian manifolds of straight lines in Euclidean space. In this chapter, we introduce a new geometric construction called a connection, which is an essential tool for defining geodesics.

Problems

• 4-1. Let $M\subseteq \mathbb {R}^n$ be an embedded submanifold and $Y\in \mathfrak X(M)$. For every point $p\in M$ and vector $v\in T_pM$, define $\nabla ^\top _v Y$ by (4.4).

  • (a) Show that $\nabla ^\top _v Y$ does not depend on the choice of extension $\tilde{Y}$ of Y.

    [Hint: Use Prop. A.28.]

  • (b) Show that $\nabla ^\top _v Y$ is invariant under rigid motions of $\mathbb {R}^n$, in the following sense: if $F\in {{\,\mathrm{E}\,}}(n)$ and $\tilde{M} = F(M)$, then $dF_p \big (\nabla ^\top _v Y\big ) = \nabla ^\top _{dF_p (v)} (F_* Y)$.

    (Used on pp. 87, 93.)

• 4-2. In your study of smooth manifolds, you have already seen another way of taking “directional derivatives of vector fields,” the Lie derivative $\mathscr {L}_XY$ (which is equal to the Lie bracket [X, Y]; see Prop. A.46). Suppose M is a smooth manifold of positive dimension.

  • (a) Show that the map $\mathscr {L}:\mathfrak X(M)\times \mathfrak X(M)\mathrel {\rightarrow }\mathfrak X(M)$ is not a connection.

  • (b) Show that there are smooth vector fields X and Y on $\mathbb {R}^2$ such that $X = Y=\partial _1$ along the $x^1$-axis, but the Lie derivatives $\mathscr {L}_X(\partial _2)$ and $\mathscr {L}_Y(\partial _2)$ are not equal on the $x^1$-axis.

• 4-3. Prove Proposition 4.7 (the transformation law for the connection coefficients).

• 4-4. Prove Theorem 4.14 (characterizing the space of connections).

• 4-5. Prove Proposition 4.16 (local formulas for covariant derivatives of tensor fields).

• 4-6. Let M be a smooth manifold and let $\nabla$ be a connection in TM. Define a map $\tau :\mathfrak X(M)\times \mathfrak X(M)\mathrel {\rightarrow }\mathfrak X(M)$ by

  $$\begin{aligned} \tau (X,Y)= \nabla _X Y - \nabla _Y X - [X, Y]. \end{aligned}$$

  • (a) Show that $\tau$ is a (1, 2)-tensor field, called the torsion tensor of $\varvec{\nabla }$.

  • (b) We say that $\nabla$ is symmetric if its torsion vanishes identically. Show that $\nabla$ is symmetric if and only if its connection coefficients with respect to every coordinate frame are symmetric: $\Gamma _{ij}^k=\Gamma _{ji}^k$. [Warning: They might not be symmetric with respect to other frames.]

  • (c) Show that $\nabla$ is symmetric if and only if the covariant Hessian $\nabla ^2 u$ of every smooth function $u\in C^\infty (M)$ is a symmetric 2-tensor field. (See Example 4.22.)

  • (d) Show that the Euclidean connection $\bar{\nabla }$ on $\mathbb {R}^n$ is symmetric.

    (Used on pp. 113, 121, 123.)

• 4-7. Let $\gamma :(-\pi ,\pi )\mathrel {\rightarrow }\mathbb {R}^2$ be the figure eight curve defined in Example 4.23. Prove that $\gamma$ is an injective smooth immersion, but its velocity vector field is not extendible.

• 4-8. Suppose M is a smooth manifold (without boundary), $I\subseteq \mathbb {R}$ is an interval (bounded or not, with or without endpoints), and $\gamma :I\mathrel {\rightarrow }M$ is a smooth curve.

  • (a) Show that for every $t_0\in I$ such that $\gamma '(t_0)\ne 0$, there is a connected neighborhood J of $t_0$ in I such that every smooth vector field along $\gamma |_J$ is extendible.

  • (b) Show that if I is an open interval or a compact interval and $\gamma$ is a smooth embedding, then every smooth vector field along $\gamma$ is extendible.

• 4-9. Let M be a smooth manifold, and let $\nabla ^0$ and $\nabla ^1$ be two connections on TM.

  • (a) Show that $\nabla ^0$ and $\nabla ^1$ have the same torsion (Problem 4-6) if and only if their difference tensor is symmetric, i.e., $D(X,Y)=D(Y, X)$ for all X and Y.

  • (b) Show that $\nabla ^0$ and $\nabla ^1$ determine the same geodesics if and only if their difference tensor is antisymmetric, i.e., $D(X,Y)=-D(Y, X)$ for all X and Y. (Used on p. 145.)

• 4-10. Suppose M is a smooth manifold endowed with a connection, $\gamma :I\mathrel {\rightarrow }M$ is a smooth curve, and $Y\in \mathfrak X(\gamma )$. Prove that if Y is parallel along $\gamma$, then it is parallel along every reparametrization of $\gamma$.

• 4-11. Suppose G is a Lie group.

  • (a) Show that there is a unique connection $\nabla$ in TG with the property that every left-invariant vector field is parallel.

  • (b) Show that the torsion tensor of $\nabla$ (Problem 4-6) is zero if and only if G is abelian.

• 4-12. Prove Proposition 4.36 (a vector or tensor field A is parallel if and only if $\nabla A \equiv 0$).

• 4-13. Prove Proposition 4.38 (properties of pullback connections).

• 4-14. Let M be a smooth n-manifold and $\nabla$ a connection in TM, let $(E_i)$ be a local frame on some open subset $U\subseteq M$, and let $\big (\varepsilon ^i\big )$ be the dual coframe.

  • (a) Show that there is a uniquely determined $n\times n$ matrix of smooth 1-forms $\big (\omega _i{}^j\big )$ on U, called the connection $\varvec{1}$ -forms for this frame, such that

    $$\begin{aligned} \nabla _{X} E_i = \omega _i{}^j (X) E_j \end{aligned}$$

    for all $X\in \mathfrak X(U)$.

  • (b) Cartan’s first structure equation: Prove that these forms satisfy the following equation, due to Élie Cartan:

    $$\begin{aligned} d\varepsilon ^j = \varepsilon ^i \wedge \omega _i{}^j + \tau ^j, \end{aligned}$$

    where $\tau ^1,\dots ,\tau ^n\in \Omega ^2(M)$ are the torsion $\varvec{2}$ -forms, defined in terms of the torsion tensor $\tau$ (Problem 4-6) and the frame $(E_i)$ by

    $$\begin{aligned} \tau (X,Y) = \tau ^j(X, Y)E_j. \end{aligned}$$

    (Used on pp. 145, 222.)