1.2.1 Embedded submanifolds

Arguably the simplest example of a two-dimensional manifold is the sphere ${\mathbb{S}}^2$. Relating to the previous example, when embedded in ${\mathbb{R}}^3$, we can view it as an idealized model for the surface of the earth. The sphere with radius 1 can be described as the set of unit vectors in ${\mathbb{R}}^3$, that is, the set

${\mathbb{S}}^2=\{(x^1,x^2,x^3)\in {\mathbb{R}}^3|{(x^1)}^2+{(x^2)}^2+{(x^3)}^2=1\}.$

Notice from the definition of the set that all points of ${\mathbb{S}}^2$ satisfy the equation ${(x^1)}^2+{(x^2)}^2+{(x^3)}^2-1=0$. We can generalize this way of constructing a manifold to the following definition.

The map F is said to give an implicit representation of the manifold. In the previous example, we used the definition with $F(x)={(x^1)}^2+{(x^2)}^2+{(x^3)}^2-1$ (see Fig. 1.1).

The fact that $\mathcal{M}=F^{-1}(0)$ is a manifold is often taken as the consequence of the submersion level set theorem instead of a definition. The theorem states that with the above assumptions, $\mathcal{M}$ has a manifold structure as constructed with charts and atlases. In addition, the topological and differentiable structure of M is in a certain way compatible with that of ${\mathbb{R}}^k$ letting us denote M as embedded in ${\mathbb{R}}^k$. For now, we will be somewhat relaxed about the details and use the construction as a working definition of what we think of as a manifold.

The map F can be seen as a set of m constraints that points in $\mathcal{M}$ must satisfy. The Jacobian matrix $dF(x)$ at a point in $x\in \mathcal{M}$ linearizes the constraints around x, and its rank $k-d$ indicates how many of them are linearly independent. In addition to the unit length constraints of vectors in ${\mathbb{R}}^3$ defining ${\mathbb{S}}^2$, additional examples of commonly occurring manifolds that we will see in this book arise directly from embedded manifolds or as quotients of embedded manifolds.

1.2.2 Charts and local euclideaness

We now describe how charts, local parameterizations of the manifold, can be constructed from the implicit representation above. We will use this to give a more abstract definition of a differentiable manifold.

When navigating the surface of the earth, we seldom use curved representations of the surface but instead rely on charts that give a flat, 2D representation of regions limited in extent. It turns out that this analogy can be extended to embed manifolds with a rigorous mathematical formulation.

The definition exactly captures the informal idea of representing a local part of the surface, the open set U, with a mapping to a Euclidean space, in the surface case ${\mathbb{R}}^2$ (see Fig. 1.2).

When using charts, we often say that we work in coordinates. Instead of accessing points on $\mathcal{M}$ directly, we take a chart $\varphi :U\to \tilde{U}$ and use points in $\varphi (U)\subseteq {\mathbb{R}}^d$ instead. This gives us the convenience of having a coordinate system present. However, we need to be aware that the choice of the coordinate system affects the analysis, both theoretically and computationally. When we say that we work in coordinates $x=(x^1,\dots ,x^d)$, we implicitly imply that there is a chart ϕ such that ${\varphi }^{-1}(x)$ is a point on $\mathcal{M}$.

It is a consequence of the implicit function theorem that embedded manifolds have charts. Proving it takes some work, but we can sketch the idea in the case of the implicit representation map $F:{\mathbb{R}}^k\to {\mathbb{R}}^m$ having Jacobian with full rank m. Recall the setting of the implicit function theorem (see e.g. [18]): Let $F:{\mathbb{R}}^{d+m}\to {\mathbb{R}}^m$ be continuously differentiable and write $(x,y)\in {\mathbb{R}}^{d+m}$ such that x denotes the first d coordinates and y the last m coordinates. Let $d_{y}F$ denote the last m columns of the Jacobian matrix dF, that is, the derivatives of F taken with respect to variations in y. If $d_{y}F$ has full rank m at a point $(x,y)$ where $F(x,y)=0$, then there exists an open neighborhood $\tilde{U}\subseteq {\mathbb{R}}^d$ of x and a differentiable map $g:\tilde{U}\to {\mathbb{R}}^m$ such that $F(x,g(x))=0$ for all $x\in \tilde{U}$.

The only obstruction to using the implicit function theorem directly to find charts is that we may need to rotate the coordinates on ${\mathbb{R}}^{d+m}$ to find coordinates $(x,y)$ and a submatrix $d_{y}F$ of full rank. With this in mind, the map g ensures that $F(x,g(x))=0$ for all $x\in \tilde{U}$, that is, the points $(x,g(x)),x\in \tilde{U}$ are included in $\mathcal{M}$. Setting $U=g(\tilde{U})$, we get a chart $\varphi :U\to \tilde{U}$ directly by the mapping $(x,g(x))\mapsto x$.

1.2.3 Abstract manifolds and atlases

We now use the concept of charts to define atlases as collections of charts and from this the abstract notion of a manifold.

An atlas thus ensures the existence of at least one chart covering a neighborhood of each point of $\mathcal{M}$. This allows the topological and differential structure of $\mathcal{M}$ to be given by a definition from the topology and differential structure of the image of the charts, that is, ${\mathbb{R}}^d$. Intuitively, the structure coming from the Euclidean spaces ${\mathbb{R}}^d$ is pulled back using ${\varphi }_i$ to the manifold. In order for this construction to work, we must ensure that there is no ambiguity in the structure we get if the domain of multiple charts cover a given point. The compatibility condition ensures exactly that.

Because of the implicit function theorem, embedded submanifolds in the sense of Definition 1.1 have charts and atlases. Embedded submanifolds are therefore particular examples of abstract manifolds. In fact, this goes both ways: The Whitney embedding theorem states that any d-dimensional manifold can be embedded in ${\mathbb{R}}^k$ with $k⩽2d$ so that the topology is induced by the one of the embedding space. For Riemannian manifolds defined later on, this theorem only provides a local $C^1$ embedding and not a global smooth embedding.

1.2.4 Tangent vectors and tangent space

As the name implies, derivatives lies at the core of differential geometry. The differentiable structure allows taking derivatives of curves in much the same way as the usual derivatives in Euclidean space. However, spaces of tangent vectors to curves behave somewhat differently on manifolds due to the lack of the global reference frame that the Euclidean space coordinate system gives. We here discuss derivatives of curves, tangent vectors, and tangent spaces.

Let $\gamma :[0,T]\to {\mathbb{R}}^k$ be a differentiable curve in ${\mathbb{R}}^k$ parameterized on the interval $[0,T]$. For each t, the curve derivative is

$\frac{d}{dt}\gamma (t)=\dot{\gamma }=\left(\begin{array}{c}\hfill \frac{d}{dt}{\gamma }^1(t)\hfill \\ \hfill ⋮\hfill \\ \hfill \frac{d}{dt}{\gamma }^k(t)\hfill \\ \hfill \hfill \end{array}\right).$

This tangent or velocity vector can be regarded as a vector in ${\mathbb{R}}^k$, denoted the tangent vector to γ at t. If $\mathcal{M}$ is an embedded manifold in ${\mathbb{R}}^k$ and $\gamma (t)\in \mathcal{M}$ for all $t\in [0,T]$, we can regard γ as a curve in $\mathcal{M}$. As illustrated on Fig. 1.3, the tangent vectors of γ are also tangential to $\mathcal{M}$ itself. The set of tangent vectors to all curves at $x=\gamma (t)$ span a d-dimensional affine subspace of ${\mathbb{R}}^k$ that approximates $\mathcal{M}$ to the first order at x. This affine space has an explicit realization as $x+\ker dF(x)$ where $x=\gamma (t)$ is the foot-point and $\ker dF$ denotes the kernel (null-space) of the Jacobian matrix of F. The space is called the tangent space $T_x\mathcal{M}$ of $\mathcal{M}$ at the point x. In the embedded manifold case, tangent vectors thus arise from the standard curve derivative, and tangent spaces are affine subspaces of ${\mathbb{R}}^k$.

On abstract manifolds, the definition of tangent vectors becomes somewhat more intricate. Let γ be a curve in the abstract manifold $\mathcal{M}$, and consider $t\in [0,T]$. By the covering assumption on the atlas, there exists a chart $\varphi :U\to \tilde{U}$ with $\gamma (t)\in U$. By the continuity of γ and openness of U, $\gamma (s)\in U$ for s sufficiently close to t. Now the curve $\tilde{\gamma }=\varphi \circ \gamma$ in ${\mathbb{R}}^d$ is defined for such s. Thus we can take the standard Euclidean derivative $\dot{\tilde{\gamma }}$ of $\tilde{\gamma }$. This gives a vector in ${\mathbb{R}}^d$. In the same way as we define the differentiable structure on $\mathcal{M}$ by definition to be that inherited from the charts, it would be natural to let a tangent vector of $\mathcal{M}$ be $\dot{\tilde{\gamma }}$ by definition. However, we would like to be able to define tangent vectors independently of the underlying curve. In addition, we need to ensure that the construction does not depend on the chart ϕ.

One approach is to define tangent vectors from their actions on real-valued functions on $\mathcal{M}$. Let $f:\mathcal{M}\to \mathbb{R}$ be a differentiable function. Then $f\circ \gamma$ is a function from $\mathbb{R}$ to $\mathbb{R}$ whose derivative is

$\frac{d}{dt}f\circ \gamma (t).$

This operation is clearly linear in f in the sense that $\frac{d}{dt}((\alpha f+\beta g)\circ \gamma )=\alpha \frac{d}{dt}(f\circ \gamma )+\beta \frac{d}{dt}(g\circ \gamma )$ when g is another differentiable function and $\alpha ,\beta \in \mathbb{R}$. In addition, this derivative satisfies the usual product rule for the derivative of the pointwise product $f\cdot g$ of f and g. Operators on differentiable functions satisfying these properties are called derivations, and we can define tangent vectors and tangent spaces as the set of derivations, that is, $v\in T_x\mathcal{M}$ is a tangent vector if it defines a derivation $v(f)$ on functions $f\in C^1(\mathcal{M},\mathbb{R})$. It can now be checked that the curve derivative using a chart above defines derivations. By the chain rule we can see that these derivations are independent of the chosen chart.

The construction of $T_x\mathcal{M}$ as derivations is rather abstract. In practice, it is often most convenient to just remember that there is an abstract definition and otherwise think of tangent vectors as derivatives of curves. In fact, tangent vectors and tangent spaces can also be defined without derivations using only the derivatives of curves. However, in this case, we must define a tangent vector as an equivalence class of curves because multiple curves can result in the same derivative. This construction, although in some sense more intuitive, therefore has its own complexities.

The set $\{T_x\mathcal{M}|x\in M\}$ has a structure of a differentiable manifold in itself. It is called the tangent bundle $T\mathcal{M}$. It follows that tangent vectors $v\in T_x\mathcal{M}$ for some $x\in \mathcal{M}$ are elements of $T\mathcal{M}$. $T\mathcal{M}$ is a particular case of a fiber bundle (a local product of spaces whose global topology may be more complex). We will later see other examples of fiber bundles, for example, the cotangent bundle $T^{⁎}\mathcal{M}$ and the frame bundle $F\mathcal{M}$.

A local coordinate system $x=(x^1,\dots x^d)$ coming from a chart induces a basis ${\partial }_x=({\partial }_{x^1},\dots {\partial }_{x^d})$ of the tangent space $T_x\mathcal{M}$. Therefore any $v\in T_x\mathcal{M}$ can be expressed as a linear combination of ${\partial }_{x^1},\dots {\partial }_{x^d}$. Writing $v^i$ for the ith entry of such linear combinations, we have $v={\sum }_{i=1}^d{v}^i{\partial }_{x^i}$.

Just as a Euclidean vector space V has a dual vector space $V^{⁎}$ consisting of linear functionals $\xi :V\to \mathbb{R}$, the tangent spaces $T_x\mathcal{M}$ and tangent bundle $T\mathcal{M}$ have dual spaces, the cotangent spaces $T_x^{⁎}\mathcal{M}$, and cotangent bundle $T^{⁎}\mathcal{M}$. For each x, elements of the cotangent space $T_x^{⁎}\mathcal{M}$ are linear maps from $T_x\mathcal{M}$ to $\mathbb{R}$. The coordinate basis $({\partial }_{x^1},\dots {\partial }_{x^d})$ induces a similar coordinate basis $(d{x}^1,\dots d{x}^d)$ for the cotangent space. This basis is defined from evaluation on ${\partial }_{x^i}$ by $d{x}^j({\partial }_{x^i})={\delta }_i^j$, where the delta-function ${\delta }_i^j$ is 1 if $i=j$ and 0 otherwise. The coordinates $v^i$ for tangent vectors in the coordinate basis had upper indices above. Similarly, coordinates for cotangent vectors conventionally have lower indices such that $\xi ={\xi }_{i}d{x}^i$ for $\xi \in T_x^{⁎}\mathcal{M}$ again using the Einstein summation convention. Elements of $T^{⁎}\mathcal{M}$ are called covectors. The evaluation $\xi (v)$ of a covector ξ on a vector v is sometimes written $(\xi |v)$ or $\langle \xi ,v\rangle$. Note that the latter notation with brackets is similar to the notation for inner products used later on.

1.2.5 Differentials and pushforward

The interpretation of tangent vectors as derivations allows taking derivatives of functions. If X is a vector field on $\mathcal{M}$, then we can use this pointwise to define a new function on $\mathcal{M}$ by taking derivatives at each point, that is, $X(f)(x)=X(x)(f)$ using that $X(x)$ is a tangent vector in $T_x\mathcal{M}$ and hence a derivation that acts on functions. If instead f is a map between two manifolds $f:\mathcal{M}\to \mathcal{N}$, then we get the differential $df:T\mathcal{M}\to T\mathcal{N}$ as a map between the tangent bundle of $\mathcal{M}$ and $\mathcal{N}$. In coordinates, this is $df{({\partial }_{x^i})}^j={\partial }_{x^i}{f}^j$ with $f^j$ being the jth component of f. The differential df is often denoted the pushforward of f because it uses f to map, that is, push, tangent vectors in $T\mathcal{M}$ to tangent vectors in $T\mathcal{N}$. For this reason, the pushforward notation $f_{⁎}=df$ is often used. When f is invertible, there exists a corresponding pullback operation $f^{⁎}=d{f}^{-1}$.

As a particular case, consider a map f between $\mathcal{M}$ and the manifold $\mathbb{R}$. Then $f_{⁎}=df$ is a map from $T\mathcal{M}$ to $T\mathbb{R}$. Because $\mathbb{R}$ is Euclidean, we can identify the tangent bundle with $\mathbb{R}$ itself, and we can consider df a map $T\mathcal{M}\to \mathbb{R}$. Being a derivative, $df{|}_{T_x\mathcal{M}}$ is linear for each $x\in \mathcal{M}$, and $df(x)$ is therefore a covector in $T_x^{⁎}\mathcal{M}$. Though the differential df is also a pushforward, the notation df is most often used because of its interpretation as a covector field.

1.3.1 Riemannian metric

A Riemannian metric is defined by a smoothly varying collection of scalar products ${\langle \cdot ,\cdot \rangle }_x$ on each tangent space $T_x\mathcal{M}$ at points x of the manifold. For each x, each such scalar product is a positive definite bilinear map ${\langle \cdot ,\cdot \rangle }_x:T_x\mathcal{M}\times T_x\mathcal{M}\to \mathbb{R}$; see Fig. 1.5. The inner product gives a norm ${\Vert \cdot \Vert }_x:T_x\mathcal{M}\to \mathbb{R}$ by ${\Vert v\Vert }^2={\langle v,v\rangle }_x$. In a given chart we can express the metric by a symmetric positive definite matrix $g(x)$. The ijth entry of the matrix is denoted $g_{ij}(x)$ and given by the dot product of the coordinate basis for the tangent space, $g_{ij}(x)={\langle {\partial }_{x^i},{\partial }_{x^j}\rangle }_x$. This matrix is called the local representation of the Riemannian metric in the chart x, and the dot product of two vectors v and w in $T_x\mathcal{M}$ is now in coordinates ${\langle v,w\rangle }_x=v^{\top }g(x)w=v^i{g}_{ij}(x)w^j$. The components $g^{ij}$ of the inverse $g{(x)}^{-1}$ of the metric defines a metric on covectors by ${\langle \xi ,\eta \rangle }_x={\xi }_i{g}^{ij}{\eta }_j$. Notice how the upper indices of $g^{ij}$ fit the lower indices of the covector in the Einstein summation convention. This inner product on $T_x^{⁎}\mathcal{M}$ is called a cometric.

1.3.2 Curve length and Riemannian distance

If we consider a curve $\gamma (t)$ on the manifold, then we can compute at each t its velocity vector $\dot{\gamma }(t)$ and its norm $\Vert \dot{\gamma }(t)\Vert$, the instantaneous speed. For the velocity vector, we only need the differential structure, but for the norm, we need the Riemannian metric at the point $\gamma (t)$. To compute the length of the curve, the norm is integrated along the curve:

$\mathcal{L}(\gamma )=\int {\Vert \dot{\gamma }(t)\Vert }_{\gamma (t)}dt=\int {\left({\langle \dot{\gamma }(t),\dot{\gamma }(t)\rangle }_{\gamma (t)}\right)}^{\frac{1}{2}}dt.$

The integrals here are over the domain of the curve, for example, $[0,T]$. We write ${\mathcal{L}}_a^b(\gamma )={\int }_a^b{\Vert \dot{\gamma }(t)\Vert }_{\gamma (t)}dt$ to be explicit about the integration domain. This gives the length of the curve segment from $\gamma (a)$ to $\gamma (b)$.

The distance between two points of a connected Riemannian manifold is the minimum length among the curves γ joining these points:

$\mathrm{dist}(x,y)=\underset{\gamma (0)=x,\gamma (1)=y}{\min }\mathcal{L}(\gamma ).$

The topology induced by this Riemannian distance is the original topology of the manifold: open balls constitute a basis of open sets.

The Riemannian metric is the intrinsic way of measuring length on a manifold. The extrinsic way is to consider the manifold as embedded in ${\mathbb{R}}^k$ and compute the length of a curve in $\mathcal{M}$ as for any curve in ${\mathbb{R}}^k$. In section 1.2.4 we identified the tangent spaces of an embedded manifold with affine subspaces of ${\mathbb{R}}^k$. In this case the Riemannian metric is the restriction of the dot product on ${\mathbb{R}}^k$ to the tangent space at each point of the manifold. Embedded manifolds thus inherit also their geometric structure in the form of the Riemannian metric from the embedding space.

1.3.3 Geodesics

In Riemannian manifolds, locally length-minimizing curves are called metric geodesics. The next subsection will show that these curves are also autoparallel for a specific connection, so that they are simply called geodesics in general. A curve is locally length minimizing if for all t and sufficiently small s, ${\mathcal{L}}_t^{t+s}(\gamma )=\mathrm{dist}(\gamma (t),\gamma (t+s))$. This implies that small segments of the curve realize the Riemannian distance. Finding such curves is complicated by the fact that any time-reparameterization of the curve is authorized. Thus geodesics are often defined as critical points of the energy functional $\mathcal{E}(\gamma )=\frac{1}{2}\int {\Vert \dot{\gamma }\Vert }^{2}dt$. It turns out that critical points for the energy also optimize the length functional. Moreover, they are parameterized proportionally to their arc length removing the ambiguity of the parameterization.

We now define the Christoffel symbols from the metric g by

$\mathrm{\Gamma }^k{}_{ij}=\frac{1}{2}{g}^{km}({\partial }_{x^i}{g}_{jm}+{\partial }_{x^j}{g}_{mi}-{\partial }_{x^m}{g}_{ij}).$

Using the calculus of variations, it can be shown that the geodesics satisfy the second-order differential system

${\ddot{\gamma }}^k+\mathrm{\Gamma }^k{}_{ij}{\dot{\gamma }}^i{\dot{\gamma }}^j=0.$

We will see the Christoffel symbols again in coordinate expressions for the connection below.

1.3.4 Levi-Civita connection

The fundamental theorem of Riemannian geometry states that on any Riemannian manifold, there is a unique connection which is compatible with the metric and which has the property of being torsion-free. This connection is called the Levi-Civita connection. For that choice of connection, shortest curves have zero acceleration and are thus geodesics in the sense of being “straight lines”. In the following we only consider the Levi-Civita connection unless explicitly stated.

The connection allows us to take derivatives of a vector field Y in the direction of another vector field X expressed as ${\mathrm{\nabla }}_{X}Y$. This is also denoted the covariant derivative of Y along X. The connection is linear in X and obeys the product rule in Y so that ${\mathrm{\nabla }}_X(fY)=X(f)Y+f{\mathrm{\nabla }}_{X}Y$ for a function $f:\mathcal{M}\to \mathbb{R}$ with $X(f)$ being the derivative of f in the direction of X using the interpretation of tangent vectors as derivations. In a local coordinate system we can write the connection explicitly using the Christoffel symbols by ${\mathrm{\nabla }}_{{\partial }_{x^i}}{\partial }_{x^j}=\mathrm{\Gamma }^k{}_{ij}{\partial }_{x^k}$. With vector fields X and Y having coordinates $X(x)=v^i(x){\partial }_{x^i}$ and $Y(x)=w^i(x){\partial }_{x^i}$, we can use this to compute the coordinate expression for derivatives of Y along X:

${\mathrm{\nabla }}_{X}Y={\mathrm{\nabla }}_{v^i{\partial }_{x^i}}(w^j{\partial }_{x^j})=v^i{\mathrm{\nabla }}_{{\partial }_{x^i}}(w^j{\partial }_{x^j})=v^i({\partial }_{x^i}{w}^j){\partial }_{x^j}+v^i{w}^j{\mathrm{\nabla }}_{{\partial }_{x^i}}{\partial }_{x^j}=v^i({\partial }_{x^i}{w}^j){\partial }_{x^j}+v^i{w}^j\mathrm{\Gamma }^k{}_{ij}{\partial }_{x^k}=(v^i({\partial }_{x^i}{w}^k)+v^i{w}^j\mathrm{\Gamma }^k{}_{ij}){\partial }_{x^k}=(X(w^k)+v^i{w}^j\mathrm{\Gamma }^k{}_{ij}){\partial }_{x^k}.$

Using this, the connection allows us to write the geodesic equation (1.8) as the zero acceleration constraint:

$0={\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }=(\dot{\gamma }({\dot{\gamma }}^k)+{\dot{\gamma }}^i{\dot{\gamma }}^j\mathrm{\Gamma }^k{}_{ij}){\partial }_{x^k}=({\ddot{\gamma }}^k+{\dot{\gamma }}^i{\dot{\gamma }}^j\mathrm{\Gamma }^k{}_{ij}){\partial }_{x^k}.$

The connection also defines the notion of parallel transport along curves. A vector $v\in T_{\gamma (t_0)}\mathcal{M}$ is parallel transported if it is extended to a t-dependent family of vectors with $v(t)\in T_{\gamma (t)}\mathcal{M}$ and ${\mathrm{\nabla }}_{\dot{\gamma }(t)}v(t)=0$ for each t. Parallel transport can thereby be seen as a map $P_{\gamma ,t}:T_{\gamma (t_0)}\mathcal{M}\to T_{\gamma (t)}\mathcal{M}$ linking tangent spaces. The parallel transport inherits linearity from the connection. It follows from the definition that γ is a geodesic precisely if $\dot{\gamma }(t)=P_{\gamma ,t}(\dot{\gamma }(t_0))$.

It is a fundamental consequence of curvature that parallel transport depends on the curve along which the vector is transported: With curvature, the parallel transports $P_{\gamma ,T}$ and $P_{\varphi ,T}$ along two curves γ and ϕ with the same end-points $\gamma (t_0)=\varphi (t_0)$ and $\gamma (T)=\varphi (T)$ will differ. The difference is denoted holonomy, and the holonomy of a Riemannian manifold vanishes only if $\mathcal{M}$ is flat, that is, has zero curvature.

1.3.5 Completeness

The Riemannian manifold is said to be geodesically complete if the definition domain of all geodesics can be extended to $\mathbb{R}$. This means that the manifold has neither boundary nor any singular point that we can reach in a finite time. For instance, ${\mathbb{R}}^d-\{0\}$ with the usual metric is not geodesically complete because some geodesics will hit 0 and thus stop being defined in finite time. On the other hand, ${\mathbb{R}}^d$ is geodesically complete. Other examples of complete Riemannian manifolds include compact manifolds implying that ${\mathbb{S}}^d$ is geodesically complete. This is a consequence of the Hopf–Rinow–de Rham theorem, which also states that geodesically complete manifolds are complete metric spaces with the induced distance and that there always exists at least one minimizing geodesic between any two points of the manifold, that is, a curve whose length is the distance between the two points.

From now on, we will assume that the manifold is geodesically complete. This assumption is one of the fundamental properties ensuring the well-posedness of algorithms for computing on manifolds.

1.3.6 Exponential and logarithm maps

Let x be a point of the manifold that we consider as a local reference point, and let v be a vector of the tangent space $T_x\mathcal{M}$ at that point. From the theory of second-order differential equations, it can be shown that there exists a unique geodesic ${\gamma }_{(x,v)}(t)$ starting from that point $x={\gamma }_{(x,v)}(0)$ with tangent vector $v={\dot{\gamma }}_{(x,v)}(0)$. This geodesic is first defined in a sufficiently small interval around zero, but since the manifold is assumed geodesically complete, its definition domain can be extended to $\mathbb{R}$. Thus the points ${\gamma }_{(x,v)}(t)$ are defined for each t and each $v\in T_x\mathcal{M}$. This allows us to map vectors in the tangent space to the manifold using geodesics: the vector $v\in T_x\mathcal{M}$ can be mapped to the point of the manifold that is reached after a unit time $t=1$ by the geodesic ${\gamma }_{(x,v)}(t)$ starting at x with tangent vector v. This mapping

${\mathrm{Exp}}_x:\begin{array}{ccc}\hfill T_x\mathcal{M}\hfill & \hfill ⟶\hfill & \mathcal{M}\hfill \\ \hfill v\hfill & \hfill ⟼\hfill & {\mathrm{Exp}}_x(v)={\gamma }_{(x,v)}(1)\hfill \end{array}$

is called the exponential map at point x. Straight lines passing 0 in the tangent space are transformed into geodesics passing the point x on the manifold, and the distances along these lines are conserved (Fig. 1.6).

When the manifold is geodesically complete, the exponential map is defined on the entire tangent space $T_x\mathcal{M}$, but it is generally one-to-one only locally around 0 in the tangent space corresponding to a local neighborhood of x on $\mathcal{M}$. We denote by $\overrightarrow{xy}$ or ${\mathrm{Log}}_x(y)$ the inverse of the exponential map where the inverse is defined: this is the smallest vector as measured by the Riemannian metric such that $y={\mathrm{Exp}}_x(\overrightarrow{xy})$. In this chart the geodesics going through x are represented by the lines going through the origin: ${\mathrm{Log}}_x{\gamma }_{(x,\overrightarrow{xy})}(t)=t\overrightarrow{xy}$. Moreover, the distance with respect to the base point x is preserved:

$\mathrm{dist}(x,y)=\Vert \overrightarrow{xy}\Vert =\sqrt{{\langle \overrightarrow{xy},\overrightarrow{xy}\rangle }_x}.$

Thus the exponential chart at x gives a local representation of the manifold in the tangent space at a given point. This is also called a normal coordinate system or normal chart if it is provided with an orthonormal basis. At the origin of such a chart, the metric reduces to the identity matrix, and the Christoffel symbols vanish. Note again that the exponential map is generally only invertible locally around $0\in T_x\mathcal{M}$, and ${\mathrm{Log}}_{x}y$ is therefore only locally defined, that is, for points y near x.

The exponential and logarithm maps are commonly referred to as the Exp and Log maps.

1.3.7 Cut locus

It is natural to search for the maximal domain where the exponential map is a diffeomorphism. If we follow a geodesic ${\gamma }_{(x,v)}(t)={\mathrm{Exp}}_x(tv)$ from $t=0$ to infinity, then it is either always minimizing for all t, or it is minimizing up to a time $t_0<\infty$. In this last case the point $z={\gamma }_{(x,v)}(t_0)$ is called a cut point, and the corresponding tangent vector $t_{0}v$ is called a tangential cut point. The set of all cut points of all geodesics starting from x is the cut locus $C(x)\in \mathcal{M}$, and the set of corresponding vectors is the tangential cut locus $\mathcal{C}(x)\in T_x\mathcal{M}$. Thus we have $C(x)={\mathrm{Exp}}_x(\mathcal{C}(x))$, and the maximal definition domain for the exponential chart is the domain $\mathcal{D}(x)$ containing 0 and delimited by the tangential cut locus.

It is easy to see that this domain is connected and star-shaped with respect to the origin of $T_x\mathcal{M}$. Its image by the exponential map covers the manifold except the cut locus, and the segment $[0,\overrightarrow{xy}]$ is transformed into the unique minimizing geodesic from x to y. Hence, the exponential chart has a connected and star-shaped definition domain that covers all the manifold except the cut locus $C(x)$:

$\begin{array}{ccc}\hfill \mathcal{D}(x)\subseteq {\mathbb{R}}^d\hfill & \hfill ⟷\hfill & \hfill \mathcal{M}-C(x)\hfill \\ \hfill \overrightarrow{xy}={\mathrm{Log}}_x(y)\hfill & \hfill ⟷\hfill & \hfill y={\mathrm{Exp}}_x(\overrightarrow{xy})\hfill \end{array}.$

From a computational point of view, it is often interesting to extend this representation to include the tangential cut locus. However, we have to take care of the multiple representations: Points in the cut locus where several minimizing geodesics meet are represented by several points on the tangential cut locus as the geodesics are starting with different tangent vectors (e.g. antipodal points on the sphere and rotation of π around a given axis for 3D rotations). This multiplicity problem cannot be avoided as the set of such points is dense in the cut locus.

The size of $\mathcal{D}(x)$ is quantified by the injectivity radius $\text{i}(\mathcal{M},x)=\mathrm{dist}(x,\mathcal{C}(x))$, which is the maximal radius of centered balls in $T_x\mathcal{M}$ on which the exponential map is one-to-one. The injectivity radius of the manifold $\text{i}(\mathcal{M})$ is the infimum of the injectivity over the manifold. It may be zero, in which case the manifold somehow tends toward a singularity (e.g. think of the surface $z=1/\sqrt{x^2+y^2}$ as a submanifold of ${\mathbb{R}}^3$).

1.4.1 Gradient and musical isomorphisms

Let f be a smooth function from $\mathcal{M}$ to $\mathbb{R}$. Recall that the differential $df(x)$ evaluated at the point $x\in \mathcal{M}$ is a covector in $T_x^{⁎}\mathcal{M}$. Therefore, contrary to the Euclidean situation where derivatives are often regarded as vectors, we cannot directly interpret $df(x)$ as a vector. However, thanks to the Riemannian metric, there is a canonical way to identify the linear form $df\in T_x^{⁎}\mathcal{M}$ with a unique vector $v\in T_x\mathcal{M}$. This is done by defining $v\in T_x\mathcal{M}$ to be a vector satisfying $df(w)={\langle v,w\rangle }_x$ for all vectors $w\in T_x\mathcal{M}$. This mapping corresponds to the transpose operator that is implicitly used in Euclidean spaces to transform derivatives of functions (row vectors) to column vectors. On manifolds, the Riemannian metric must be specified explicitly since the coordinate system used may not be orthonormal everywhere.

The mapping works for any covector and is often denoted the sharp map ${}^{♯}:T^{⁎}\mathcal{M}\to T\mathcal{M}$. It has an inverse in the flat map ${}^{♭}:T\mathcal{M}\to T^{⁎}\mathcal{M}$. In coordinates, ${({\xi }^{♯})}^i=g^{ij}{\xi }_j$ for a covector $\xi ={\xi }_{j}d{x}^j$, and ${(v^{♭})}_i=g_{ij}{v}^j$ for a vector $v={\partial }_{x^j}{v}^j$. The maps ${}^{♯}$ and ${}^{♭}$ are denoted musical isomorphisms because they raise or lower the indices of the coordinates.

We can use the sharp map to define the Riemannian gradient as a vector:

$\mathrm{grad}f={(df)}^{♯}.$

This definition corresponds to the classical gradient in ${\mathbb{R}}^k$ using the standard Euclidean inner product as a Riemannian metric. Using the coordinate representation of the sharp map, we get the coordinate form ${(\mathrm{grad}f)}^i=g^{ij}{\partial }_{x^j}f$ of the gradient.

1.4.2 Hessian and Taylor expansion

The covariant derivative of the gradient, the Hessian, arises from the connection ∇:

$\mathrm{Hess}f(X,Y)={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}f=({\mathrm{\nabla }}_X(df))Y=\langle {\mathrm{\nabla }}_X\mathrm{grad}f,Y\rangle .$

Here the two expressions on the right are given using the action of the connection on the differential form df (a covector) or the vector field $\mathrm{grad}f={(df)}^{♯}$. Its expression in a local coordinate system is

$\mathrm{Hess}f=\mathrm{\nabla }df=({\partial }_{x^i{x}^j}f-\mathrm{\Gamma }^k{}_{ij}{\partial }_{k}f)d{x}^{i}d{x}^j.$

Let now $f_x$ be the expression of f in a normal coordinate system at x. Its Taylor expansion around the origin in coordinates is

$f_x(v)=f_x(0)+d{f}_{x}v+\frac{1}{2}{v}^{\top }{H}_{f_x}v+O({\Vert v\Vert }^3),$

where $d{f}_x=({\partial }_{x^i}f)$ is the Jacobian matrix of first-order derivatives, and $H_{f_x}=({\partial }_{x^i{x}^j}f)$ is the Euclidean Hessian matrix. Because the coordinate system is normal, we have $f_x(v)=f({\mathrm{Exp}}_x(v))$. Moreover, the metric at the origin reduces to the identity: $d{f}_x={(\mathrm{grad}f)}^T$, and the Christoffel symbols vanish so that the matrix of second derivatives $H_{f_x}$ corresponds to the Hessian Hess f. Thus the Taylor expansion can be written in any coordinate system:

$f({\mathrm{Exp}}_x(v))=f(x)+\mathrm{grad}f(v)+\frac{1}{2}\mathrm{Hess}f(v,v)+O({\Vert v\Vert }^3).$

1.4.3 Riemannian measure or volume form

In a vector space with basis $\mathcal{A}=(a_1,\dots a_n)$ the local representation of the metric is given by $g=A^{\top }A$, where $A=[a_1,\dots a_n]$ is the matrix of coordinates change from $\mathcal{A}$ to an orthonormal basis. Similarly, the measure or the infinitesimal volume element is given by the volume of the parallelepiped spanned by the basis vectors: $d\mathcal{V}=|A|dx=\sqrt{|g|}dx$ with $|\cdot |$ denoting the matrix determinant. In a Riemannian manifold $\mathcal{M}$, the Riemannian metric $g(x)$ induces an infinitesimal volume element on each tangent space, and thus a measure on the manifold that in coordinates has the expression

$d\mathcal{M}(x)=\sqrt{|g(x)|}dx.$

The cut locus has null measure, and we can therefore integrate indifferently in $\mathcal{M}$ or in any exponential chart. If f is an integrable function of the manifold and $f_x(v)=f({\mathrm{Exp}}_x(v))$ is its image in the exponential chart at x, then we have

${\int }_{x\in \mathcal{M}}f(x)d\mathcal{M}(x)={\int }_{v\in \mathcal{D}(x)}{f}_x(v)\sqrt{|g({\mathrm{Exp}}_x(v))|}dv.$

1.4.4 Curvature

The curvature of a Riemannian manifold measures its deviance from local flatness. We often have a intuitive notion of when a surface embedded in ${\mathbb{R}}^3$ is flat or curved; for example, a linear subspace of ${\mathbb{R}}^3$ is flat, whereas the sphere ${\mathbb{S}}^2$ is curved. This idea of curvature is expressed in the Gauss curvature. However, for high-dimensional spaces, the mathematical description becomes somewhat more intricate. We will further on see several notions of curvature capturing aspects of the nonlinearity of the manifold with varying details. It is important to note that whereas vanishing curvature implies local flatness of the manifold, this is not the same as the manifold being globally Euclidean. An example is the torus ${\mathbb{T}}_2$, which can both be embedded in ${\mathbb{R}}^3$ inheriting nonzero curvature and be embedded in ${\mathbb{R}}^4$ in a way in which it inherits a flat geometry. In both cases the periodicity of the torus remains, which prevents it from being a vector space.

The curvature of a Riemannian manifold is described by the curvature tensor $R:T\mathcal{M}\times T\mathcal{M}\times T\mathcal{M}\to T\mathcal{M}$. It is defined from the covariant derivative by evaluation on vector fields X, Y, Z:

$R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z.$

The bracket $[X,Y]$ denotes the anticommutativity of the fields X and Y. If f is a differentiable function on $\mathcal{M}$, then the new vector field produced by the bracket is given by its application to f: $[X,Y]f=X(Y(f))-Y(X(f))$. The curvature tensor R can intuitively be interpreted at $x\in \mathcal{M}$ as the difference between parallel transporting the vector $Z(x)$ along an infinitesimal parallelogram with sides $X(x)$ and $Y(x)$; see Fig. 1.7 (left). As noted earlier, parallel transport is curve dependent, and the difference between transporting infinitesimally along $X(x)$ and then $Y(x)$ as opposed to along $Y(x)$ and then $X(x)$ is a vector in $T_x\mathcal{M}$. This difference can be calculated for any vector $z\in T_x\mathcal{M}$. The curvature tensor when evaluated at X, Y, that is, $R(X,Y)$, is the linear map $T_x\mathcal{M}\to T_x\mathcal{M}$ given by this difference.

The reader should note that two different sign conventions exist for the curvature tensor: definition (1.10) is used in a number of reference books in physics and mathematics [20,16,14,24,11]. Other authors use a minus sign to simplify some of the tensor notations [26,21,4,5,1] and different order conventions for the tensors subscripts and/or a minus sign in the sectional curvature defined below (see e.g. the discussion in [6, p. 399]).

The curvature can be realized in coordinates from the Christoffel symbols:

$R({\partial }_{x^i},{\partial }_{x^j}){\partial }_{x^k}=R^m{}_{kij}{\partial }_{x^m}=(\mathrm{\Gamma }^l{}_{jk}\mathrm{\Gamma }^m{}_{il}-\mathrm{\Gamma }^l{}_{ik}\mathrm{\Gamma }^m{}_{jl}+{\partial }_i\mathrm{\Gamma }^m{}_{jk}-{\partial }_j\mathrm{\Gamma }^m{}_{ik}){\partial }_{x^m}.$

The sectional curvature κ measures the Gaussian curvature of 2D submanifolds of $\mathcal{M}$, the Gaussian curvature of each point being the product of the principal curvatures of curves passing the point. The 2D manifolds arise as the geodesic spray of a 2D linear subspace of $T_x\mathcal{M}$; see Fig. 1.7 (right). Such a 2-plane can be represented by basis vectors $u,v\in T_x\mathcal{M}$, in which case the sectional curvature can be expressed using the curvature tensor by

$\kappa (u,v)=\frac{\langle R(u,v)v,u\rangle }{{\Vert u\Vert }^2{\Vert v\Vert }^2-{\langle u,v\rangle }^2}.$

The curvature tensor gives the notion of Ricci and scalar curvatures, which both provide summary information of the full tensor R. The Ricci curvature Ric is the trace over the first and last indices of R with coordinate expression

$R_{ij}=R_{kij}{}^k=R^k{}_{ikj}=g^{kl}{R}_{ikjl}.$

Taking another trace, we get the scalar valued quantity, the scalar curvature S:

$S=g^{ij}{R}_{ij}.$

Note that the cometric appears to raise one index before taking the trace.