Last time we have seen how to take the covariant derivative of a vector field with respect to another vector field, via the definition of a connection $\nabla$ on the tangent bundle $TM$ of a smooth manifold $M$. This allows us to define the acceleration of a curve, and define the notion of a geodesic as a curve with zero acceleration. However, we have also seen that there are many choices for $\nabla$, one for each choice of the Christoffel symbols $\Gamma_{ij}^k$.

In this post, we discuss the situation when $M$ is endowed with a Riemannian metric. We will see that there is a unique connection, called the Levi-Civita connection, which is compatible with the metric and satisfies a symmetry property. This will allow us to define Riemannian geodesics with nice naturality properties, and also leads to the exponential map, which encodes the collective behavior of geodesics.

As before, this is following Lee’s Riemannian Manifold, $\S3$ and $\S5$.


Riemannian metric

Let $(M,\g)$ be a Riemannian manifold, which means $M$ is a smooth manifold and $\g$ is a Riemannian metric, which is a $2$-tensor field, so at each $p \in M$, we have a function that takes two tangent vectors:

which is symmetric, which means $\g(u,v) = \g(v,u)$, and positive definite, which means $\g(v,v) > 0$ if $v \neq 0$. A Riemannian metric $\g$ thus defines an inner product on each tangent space $T_pM$:

Given a coordinate chart $(x^i) \equiv (x^1, \dots, x^n)$ on a neighborhood of $p$, we can write $\g$ locally as:

where the coefficient matrix $\g_{ij} = \langle \partial_i, \partial_j \rangle$ is symmetric in $i$ and $j$ and depends smoothly on $p \in M$, $\partial_i \equiv \partial / \partial x^i$ is the induced local coordinate frame on $T_pM$, and and $(dx^i)$ is the dual coframe on $T_p^\ast M$, defined by $dx^i(\partial_j) = \delta^i_j$. Since $\g_{ij}$ is symmetric, we can also write the above as:

where $dx^i dx^j := \frac{1}{2}(dx^i \otimes dx^j + dx^j \otimes dx^i)$ is the symmetric product (if we think of $dx^i$ as a column vector $e_i$, then the matrix $dx^i \otimes dx^j = e_i e_j^\top$ is not symmetric, but $dx^i dx^j = \frac{1}{2} (e_i e_j^\top + e_j e_i^\top)$ is).

For example, when $M = \R^n$, the Euclidean metric $\bar \g$ is given by $\bar \g_{ij} = \delta_{ij}$, so $\bar \g = \sum_i dx^i dx^i$.

Let $(M,\g)$ and $(\tilde M, \tilde \g)$ be Riemannian manifolds. Recall that given a diffeomorphism $\varphi \colon M \to \tilde M$, we can define the pullback metric $\varphi^\ast \tilde \g$ on $M$ by pushing forward tangent vectors from $TM$ to $T \tilde M$ using the differential of $\varphi$, and then evaluating the metric $\tilde \g$ on $\tilde M$:

We say $\varphi$ is an isometry if this pullback metric agrees with the underlying metric on $M$, i.e., if $\g = \varphi^\ast \tilde \g$. Isometries are the isomorphisms in the category of Riemannian manifolds, and as Lee explains (p. 24), “Riemannian geometry is concerned primarily with properties that are preserved by isometries.”


Tangential connection

The motivation for the definition of the Levi-Civita connection comes from the tangential connection. The setting is as follows. Suppose $M \subset \R^n$ is an embedded submanifold with the induced metric $\g$, which means the inclusion map $\iota \colon M \hookrightarrow \R^n$ is an embedding (a homeomorphism onto its image with the subspace topology), and $\g = \iota^\ast \bar \g$ where $\bar \g$ is the Euclidean metric on $\R^n$. By the Nash embedding theorem, any Riemannian metric on any manifold can be realized this way (which is quite remarkable).

We define the tangential connection

by setting

where the vector fields $v, w$ are extended arbitrarily to $\R^n$, $\bar \nabla$ is the Euclidean connection on $\R^n$, and for any point $p \in M$, $\pi^\top \colon T_p\R^n \to T_pM$ is the orthogonal projection.

In some sense, $\nabla^\top$ is the natural connection to use when $M$ is an embedded submanifold of $\R^n$. In particular, it has the following nice properties. First, observe that the Euclidean connection $\bar \nabla$ satisfies the product rule with respect to the Euclidean metric:

which follows easily by expanding in components. Thus, the same product rule also holds for the tangential component $\nabla^\top$, by restricting the vector fields to be tangent to $M$ and interpreting the inner products as with respect to the induced metric on $M$. This is the compatibility of $\nabla^\top$ with the metric.

Second, observe that the Euclidean connection is symmetric:

where recall $[v,w] = vw - wv$ is the Lie bracket of $v$ and $w$, which measures the degree of noncommutativity of $v$ and $w$. Indeed, in this case $\bar \nabla_v w$ is just $vw$, and similarly $\bar \nabla_w v$ is equal to $wv$. Then by restricting the vector fields to be tangent to $M$, we find that the tangential connection $\nabla^\top$ is also symmetric. It turns out these two properties are enough to characterize a unique connection.


Levi-Civita connection

The properties above motivate the following definition. Let $(M,\g)$ be a Riemannian manifold. We say a linear connection $\nabla$ is compatible with $\g$ if it satisfies the following product rule:

for all vector fields $u,v,w$. We can show (Lee, Lemma 5.2) that the following are equivalent:

  • $\nabla$ is compatible with $\g$.

  • $\nabla \g \equiv 0$

  • For any vector fields along any curve $\gamma$,

  • If $v$ and $w$ are parallel vector fields along a curve $\gamma$ (which recall means $D_t v \equiv 0$ and $D_t w \equiv 0$), then $\langle v,w \rangle$ is constant.

  • Parallel translation $P_{t_0t_1}\colon T_{\gamma(t_0)}M \to T_{\gamma(t_1)}M$ is an isometry for each $t_0$, $t_1$. Recall that given $v_0 \in T_{\gamma(t_0)}M$, there is a unique parallel vector field $v$ along $\gamma$ with $v(t_0) = v_0$, and $P_{t_0t_1} (v_0)$ is defined to be $v(t_1)$.

Next, we define the torsion tensor, which is a $\binom{2}{1}$-tensor field $\tau \colon \T(M) \times \T(M) \to \T(M)$, by:

where $[v,w] = vw - wv$ is the Lie bracket of $v$ and $w$. We say that $\nabla$ is symmetric if its torsion tensor vanishes, that is:

Then we have the following Fundamental Lemma of Riemannian Geometry (Lee, Theorem 5.4):

Theorem: Let $(M,\g)$ be a Riemannian (or pseudo-Riemannian) manifold. There exists a unique linear connection $\nabla$ on $M$ that is compatible with $\g$ and symmetric.

This connection is called the Levi-Civita connection (also called Riemannian connection in Lee’s book).

To prove uniqueness, we can derive a formula for $\nabla$, which we can also use to construct existence. In terms of the local coordinate frame $(\partial_i)$, whose Lie brackets are zero, the properties above give us:

Using the definition of the metric coefficients $\g_{ij} = \langle \partial_i, \partial_j \rangle$ and the Christoffel symbols $\nabla_{\partial_i} \partial_j = \Gamma_{ij}^m \partial_m$, we obtain:

Then multiplying both sides by the inverse matrix $g^{lk}$ and noting that $g_{ml}g^{lk} = \delta_m^k$, we get the formula for the Christoffel symbols:

Conversely, this formula defines a connection $\nabla$, which is symmetric since $\Gamma_{ij}^k = \Gamma_{ji}^k$. Furthermore, it is compatible with the metric, since the components of $\nabla \g$ are

by the definition of the Christoffel symbols.


Riemannian geodesics

Armed with the definition of the Levi-Civita connection above, we define Riemannian geodesics (or simply geodesics) to be geodesic curves with respect to the Levi-Civita connection. We can substitute the formula for $\Gamma_{ij}^k$ above to the geodesic equation to obtain a system of differential equations involving the metric, which in some nice cases we can solve explicitly.

Riemannian geodesics have many nice properties. For example, all Riemannian geodesics are constant speed curves (Lee, Lemma 5.5), which means $|\dot \gamma(t)|$ is independent of $t$. This is because $\dot \gamma$ is parallel along $\gamma$, since $\gamma$ is a geodesic which means $D_t \dot \gamma = \nabla_{\dot \gamma} \dot \gamma = 0$, so by the fourth property above, $|\dot \gamma(t)| = \langle \dot \gamma, \dot \gamma \rangle^{1/2}$ is constant. Furthermore, Riemannian geodesics are locally minimizing, which is of primary interest in many applications, and often taken to be the defining property of Riemannian geodesics.

However, as Lee explains, a deeper reason is naturality. Because the Levi-Civita connections (and hence the Riemannian geodesics) are defined in coordinate-invariant terms, they behave well with respect to isometries. Concretely, let $\varphi \colon (M,\g) \to (\tilde M, \tilde \g)$ be an isometry. Then (Lee, Proposition 5.6):

  • $\varphi$ takes the Levi-Civita connection $\nabla$ of $\g$ to the Levi-Civita connection $\tilde \nabla$ of $\tilde \g$, in the sense that

  • If $\gamma$ is a curve in $M$ and $v$ is a vector field along $\gamma$, then

  • $\varphi$ takes geodesics to geodesics: if $\gamma$ is the geodesic in $M$ with initial point $p$ and initial velocity $v$, then $\varphi \circ \gamma$ is the geodesic in $\tilde M$ with initial point $\varphi(p)$ and initial velocity $\varphi_\ast v$.

Indeed, as Lee explains about the Levi-Civita connection (p. 71):

“The real reason why this connection has been anointed as “the” Riemannian connection is this: symmetry and compatibility are invariantly-defined and natural properties that force the connection to coincide with the tangential connection whenever $M$ is realized as a submanifold of $\R^n$ with the induced metric (which the Nash embedding theorem guarantees is always possible).”


Exponential map

The exponential map is a map from the tangent bundle into the manifold that encodes the dependence of the geodesics on the initial data. Recall that any initial point $p \in M$ and any initial velocity vector $v \in T_pM$ determines a unique maximal geodesic $\gamma_v$.

We define the exponential map $\exp \colon \E \to M$ by

where the domain $\E$ is given by

For each $p \in M$, the restricted exponential map $\exp_p$ is the restriction of $\exp$ to $\E_p := \E \cap T_pM$.

We can show (Lee, Proposition 5.7) that $\E$ is an open subset of $TM$ containing the zero section, and each set $\E_p := \E \cap T_pM$ is star-shaped with respect to $0$. We can also show that the exponential map is smooth. Moreover, for each $v \in TM$, the geodesic $\gamma_v$ is given by

for all $t$ such that either side is defined. We also have the rescaling property:

for any $c,t \in \R$ whenever either side is defined.

The proof of the smoothness of the exponential map (Lee, p.74) is pretty interesting. It involves the following vector field $G$ on $\pi^{-1}(U) \subset TM$, with respect to the standard coordinates $(x^i,v^i)$ induced by local coordinates $(x^i)$ on an open set $U \subset M$:

so that the integral curves of $G$ are precisely the geodesic equation, under the substitution $v^k = \dot x^k$. As Lee explains, “the importance of $G$ stems from the fact that it actually extends to a global vector field on the total space of the tangent bundle $TM$, called the geodesic vector field.” The key observation is that for any $f \in C^\infty(TM)$, $G$ acts on $f$ by:

so it is taking the partial derivative along the geodesic, now expressed in coordinate-free language.

Finally, the naturality of the Levi-Civita connection gives rise to the naturality of the exponential map (Lee, Proposition 5.9), which means if $\varphi \colon (M,\g) \to (\tilde M, \tilde \g)$ is an isometry, then for any $p \in M$, the following diagram commutes: