Tangent Spaces and Riemannian Metrics

A tangent vector describes a direction in which a function can be differentiated at a point. In Euclidean space this is an ordinary directional derivative. On a manifold, a chart gives the same description locally, and the chain rule tells us how the components change when we choose another chart.

We use this transformation rule to construct the tangent bundle and the differential of a smooth map. A Riemannian metric then adds lengths and angles by choosing an inner product in each tangent space, smoothly in the base point.

1. \(\mathbb R^d\)

Begin in an open Euclidean set, where tangent vectors and directional derivatives can be identified explicitly. Let \(x \in \mathbb R^d\), \(\Omega \subset \mathbb R^d\) be an open set, and fix \(x_0 \in \Omega\). Recall the usual definition of the tangent space of \(\Omega\) at \(x_0\): we identify

\[T_{x_0} \Omega \cong \{x_0\} \times E ~~,~~ \text{where } E = \operatorname{span} \left( \left\{ \dpd[]{}{x_1}, \cdots, \dpd[]{}{x_d} \right\} \right) ~.\]

If \(\Omega, \Omega'\subset \mathbb R^d\) are open, then, if \(f: \Omega \to \Omega'\) is differentiable, we can define the derivative \(\dif f(x_0)\) at \(x_0 \in \Omega\) as the induced linear map between tangent spaces (under chosen coordinate systems), namely,

\[\begin{aligned} \dif f (x_0) &:& T_{x_0} \Omega &\to&& T_{f(x_0)} \Omega' \\ && v^i \dpd{}{{x^i}} = v &\mapsto&& v^i \dpd{{f^j}}{{x^i}} (x_0) \dpd{}{{f^j}} \end{aligned}\]

where the Einstein summation convention is used above. Via this calculation, we have the (formal) formula for mapping between tangent spaces, and we will use the shorthand notation

\[\dif f(x) (v) = v^i \dpd{{f^j}}{{x^i}} (x)\]

hereafter.

By definition, we have \(T \Omega = \Omega \times E \subset \Omega \times \mathbb R^d\). It follows that \(T \Omega\) is an open subset of \(\mathbb R^d \times \mathbb R^d\), i.e. it is a differentiable manifold.

Definition 1. For an open set \(\Omega\subset\mathbb R^d\), its tangent bundle is the disjoint union \(T\Omega=\bigsqcup_{p\in\Omega}T_p\Omega\), identified with \(\Omega\times\mathbb R^d\) by \((p,v)\). The base point is part of the data: vectors in different fibres belong to different tangent spaces.

2. Manifolds

Now, we can speak about what it means to differentiate on a differentiable manifold \(M\) of dimension \(d\). Let \(U \subset M\) be an open set, and take \(p \in U\). Let \(\mathrm x : U \to \mathbb R^d\) be a chart. It follows that we can make the following identification via the chart \(\rm x\):

\[\begin{aligned} T_p M \cong T_{\mathrm x(p)} \mathrm x(U) ~. \end{aligned}\]

Remark 2 (Identifying tangent vectors across charts). At \(p\in U\cap U'\), let \(x:U\to\mathbb R^d\) and \(x':U'\to\mathbb R^d\) be charts. Coordinate vectors \(v\) and \(v'\) represent the same tangent vector when

\[v'=D(x'\circ x^{-1})_{x(p)}v.\]

This is a linear isomorphism, since differentiating the inverse transition gives the inverse matrix. The chain rule makes the identification transitive through a third chart; the identity transition and inverse transition give reflexivity and symmetry. Thus equivalence classes of coordinate vectors give a well-defined vector space \(T_pM\) of dimension \(d\).

Remark 3 (Directional derivatives). If \(w\in T_pM\) is represented by \(v\) in chart \(x\), define

\[w(f)=D(f\circ x^{-1})_{x(p)}v\]

for smooth functions near \(p\). A change of chart contributes the inverse matrix in the differential of \(f\) and the forward matrix in \(v\), which cancel by the chain rule. Hence this definition is independent of the chart. It is linear in \(f\) and satisfies \(w(fh)=f(p)w(h)+h(p)w(f)\). The differential \(df_p\) is the linear map \(T_pM\to T_{f(p)}\mathbb R\cong\mathbb R\) given by \(w\mapsto w(f)\), not a map \(M\to\mathbb R\).

Let \(F: M \to N\) be a differentiable map between differentiable manifolds of dimension \(d\) and \(e\) for \(M\) and \(N\), respectively. Let \(\dif F\) denote the differential of \(F\) in local charts. For this, let \(\mathrm x : U \subset M \to \mathbb R^d\) and \(\mathrm y : V \subset N \to \mathbb R^e\), so we have the formula

\[\begin{aligned} \dif F = \dif\, (y \circ F \circ x^{-1}) : \mathbb R^d \to \mathbb R^e ~. \end{aligned}\]

The differential is represented in coordinates by the Jacobian matrix. Let \((x^1,\cdots,x^d)\) be coordinates in \(U\), and \((y^1,\cdots,y^e)\) be those in \(V\). As such, we have

\[F(x) = (F^1 (x^1,\cdots,x^d), \cdots, F^e (x^1,\cdots,x^d))\]

in local coordinates. By preceding arguments, we can define \(\dif F (x) : T_x M \to T_{F(x)} N\) as an induced linear map between vector spaces. Furthermore, we can represent the transformation (between finite-dimensional Euclidean spaces!) using the following matrix:

\[\left( \dpd{{F^{\alpha}}}{{x^{i}}} \right)_{\alpha \in \llbracket e \rrbracket, i \in \llbracket d \rrbracket} ~.\]

Clearly, on manifolds, we can also speak about base changes between tangent spaces. Suppose, under \(F\),

\[\begin{aligned} (x^1,\cdots,x^d) &\mapsto (\psi^1,\cdots,\psi^d) ~, \\ (F^1,\cdots,F^e) &\mapsto (\varphi^1,\cdots,\varphi^e) ~. \end{aligned}\]

Then, \(\dif F\) in new coordinates can be represented by the following change of basis matrix:

\[ \dpd{{\varphi^\beta}}{{\psi^j}} = \dpd{{\varphi^\beta}}{{F^\alpha}} \dpd{{F^\alpha}}{{x^i}} \dpd{{x^i}}{{\psi^j}} ~.\tag{1}\]

With the understanding of (1), we can speak about tangent bundles on manifolds, which are defined up to charts.

Definition 4. The tangent bundle is \(TM=\bigsqcup_{p\in M}T_pM\). Concretely, take triples \((p,x,v)\), where \(x:U\to\mathbb R^d\) is a chart with \(p\in U\) and \(v\in\mathbb R^d\), and identify two triples with the same base point using the derivative transition in Remark 2. The projection sends the equivalence class of \((p,x,v)\) to \(p\). Merely listing chart-vector pairs without this identification would count the same tangent vector many times.

It follows that

\[\Pi : T M \to M : (p, v) \mapsto p\]

is a natural projection map we can define. The question is when if \(\Pi\) a smooth map between \(TM\) and \(M\). We can answer that in the positive.

Theorem 5. Let \(M\) be a differentiable manifold of dimension \(d\). Then, \(TM\) admits the natural topology and structure of \(2d\)-dimensional manifold, and \(\Pi: TM \to M\) is smooth.

Proof. For each chart \(x:U\to\mathbb R^d\), define

\[\widetilde x:\Pi^{-1}(U)\longrightarrow x(U)\times\mathbb R^d,\qquad \widetilde x(p,w)=(x(p),v),\]

where \(v\) is the coordinate representative of \(w\). The equivalence relation makes this a bijection. On an overlap with chart \(y\), put \(F=y\circ x^{-1}\). The induced transition is

\[\widetilde y\circ\widetilde x^{-1}(u,v)=(F(u),DF_u v).\]

It is smooth, and its inverse is the same expression for \(F^{-1}\). Declare the sets \(\widetilde x^{-1}(O)\), for Euclidean-open \(O\), to be a basis; compatibility of the transitions makes each \(\widetilde x\) a homeomorphism onto its open image.

For Hausdorffness, vectors with distinct base points are separated by inverse images of disjoint base neighbourhoods; vectors in the same fibre lie in one bundle chart and are separated by disjoint coordinate neighbourhoods there. A second-countable manifold has a countable subatlas: choose a countable subcover of any atlas using a countable basis. Rational balls in each of the resulting countably many bundle charts provide a countable basis for \(TM\). Thus this atlas gives a \(2d\)-dimensional smooth manifold, not just locally compatible coordinates. In these charts \(\Pi\) is \((u,v)\mapsto u\), so it is smooth. ◻

Proposition 6. Let \(M\) be a differentiable manifold of dimension \(d\), and suppose \(M\) can be covered by a single smooth chart. Then, \(T M\) is diffeomorphic to \(M \times \mathbb R^d\).

Proof. Let \(x:M\to\Omega\subset\mathbb R^d\) be the global chart. Define

\[\Theta:TM\longrightarrow M\times\mathbb R^d,\qquad \Theta(p,w)=(p,dx_p(w)).\]

Its inverse sends \((p,v)\) to \((p,d(x^{-1})_{x(p)}v)\). Both maps are smooth in the bundle coordinates of Theorem 5, and their composites are the identity by the chain rule. Thus \(TM\cong M\times\mathbb R^d\). The intermediate coordinate space is \(\Omega\times\mathbb R^d\); composing its first factor with \(x^{-1}\) is what produces the asserted target. ◻

3. Riemannian Metric

The smooth structure gives each tangent space a vector space structure, but it does not choose an inner product. To measure tangent vectors, we make this additional choice smoothly at every point. This is the role of a Riemannian metric.

Definition 7. A Riemannian metric is a symmetric inner product defined on the tangent space. Specifically, \(g : T_p M \times T_p M \to \mathbb R\) is said to be a Riemannian metric if \(g\) depends smoothly on \(p \in M\).

In a chart on a \(d\)-dimensional manifold, a Riemannian metric has smooth coefficients \(g_{ij}(p)=g_p(\partial_i,\partial_j)\), with \(g_{ij}=g_{ji}\) and

\[\sum_{i,j=1}^d g_{ij}(p)\xi^i\xi^j>0\qquad\text{for every nonzero }\xi\in\mathbb R^d.\]

Positive definiteness is a condition on all nonzero vectors, not on pairs of distinct indices. The coordinate-change law below shows that smoothness in one chart implies smoothness in every compatible chart.

Now, it makes sense to discuss how the tangent vectors \(v, w \in T_p M\) relate to each other. Let \((v^1,\cdots,v^d)\) and \((w^1,\cdots,w^d)\) be coordinate representations of \(v\) and \(w\), respectively. So, we can define an inner product

\[\langle v, w \rangle = g_{ij} (x(p)) v^i w^j\]

where \(v = v^i \pd{}{{x^i}}\) and \(w = w^j \pd{}{{x^j}}\). In particular, this means \(\left\langle \pd{}{{x^i}}, \pd{}{{x^j}} \right\rangle = g_{ij}\). This implies we have an induced norm

\[\lVert v \rVert \triangleq \sqrt{\langle v, v \rangle} ~.\]

We can investigate the change in coordinates behaviour under a map \(f\). Let \(y = f(x)\) denote the new coordinates. Accordingly, we have \(v = (\widetilde v^1, \cdots, \widetilde v^d)\) and \(w = (\widetilde w^1, \cdots, \widetilde w^d)\), where \(\widetilde v^j = v^i \pd{{f^j}}{{x^i}}\) and \(\widetilde w^j = w^i \pd{{f^j}}{{x^i}}\). Furthermore, the following equivalence holds:

\[\begin{aligned} \langle v, w \rangle = h_{k\ell}(f(x)) \widetilde v^k \widetilde w^\ell = g_{ij} (x) v^i w^j ~. \end{aligned}\]

This suggests

\[\begin{aligned} g_{ij} (x) = h_{k\ell} (f(x)) \dpd{{f^k}}{{x^i}} \dpd{{f^\ell}}{{x^j}} \end{aligned}\]

for all \(v, w \in T_p M\).

Example 8. The Euclidean metric is defined via \(\langle v, w \rangle = \delta_{ij} v^i w^j\).

We will use \(g(\dot,\dot)\) and \(\langle \cdot, \cdot \rangle\) to denote the Riemannian metric interchangeably. The next construction uses partitions of unity to establish the existence of a Riemannian metric on differentiable manifolds.

Proposition 9. Every differentiable manifold can be equipped with a Riemannian metric.

Proof. On each chart \(x_\alpha:U_\alpha\to\mathbb R^d\), pull back the Euclidean metric:

\[(g_\alpha)_p(v,w)=\langle (dx_\alpha)_pv,(dx_\alpha)_pw\rangle_{\mathbb R^d} =\sum_{i=1}^d dx_\alpha^i(v)\,dx_\alpha^i(w).\]

The differential of a chart is an isomorphism, so this form is positive definite. Let \(\{p_\alpha\}\) be a smooth partition of unity with locally finite supports and \(\operatorname{spt}p_\alpha\subset U_\alpha\). Extend \(p_\alpha g_\alpha\) by zero off \(U_\alpha\). This extension is smooth: every point outside the support has a neighbourhood where \(p_\alpha\) vanishes, including points on the boundary of the chart.

Define \(g=\sum_\alpha p_\alpha g_\alpha\). Locally this is a finite sum of smooth symmetric bilinear forms. If \(v\in T_pM\) is nonzero, at least one \(p_\alpha(p)>0\), and for that index \((g_\alpha)_p(v,v)>0\); all other terms are nonnegative. Hence \(g_p(v,v)>0\). This proves smoothness, symmetry, bilinearity and positive definiteness. The zero vector has value zero, as required. ◻

Note that the construction in the proof of Proposition 9 says nothing about uniqueness; there are, in general, many Riemannian metrics on a differentiable manifold. By way of nomenclature, the Riemannian metric is the first fundamental form of a differentiable manifold \(M\), i.e. an inner product on the tangent space. The second fundamental form, which measures the bending of an immersed submanifold, will be the subject of Section The Second Fundamental Form and the Gauss Equations.

Remark 10. Regarding notation of the Riemannian metric: we use \(g\) and \(\langle\cdot,\cdot\rangle\) interchangeably. Whenever we need to invoke the metric coefficients, we always use \(g\) to denote the metric. Elsewhere, \(\langle\cdot,\cdot\rangle\) is used.

Reading: Carmo (1992), Riemannian geometry.