Riemannian Metrics

Riemannian Metrics

In [Manifolds] §Differential Forms, we used the exterior algebra to define the exterior algebra bundle

\[\bigwedge\nolimits(T^\ast M)\cong\bigoplus_{k=0}^n\bigwedge\nolimits^k(T^\ast M)\]

and defined a differential form as a smooth section of this bundle. A similar construction can be carried out using the symmetric algebra, and unlike the case of the exterior algebra, the case \(k=2\) is of particular interest. This is because when \(k=2\), the resulting elements of \(\mathcal{S}^2(T^\ast M)\) define symmetric bilinear forms on \(TM\).

Fix a point \(p\in M\). Then for a smooth section \(g\in\Gamma(\mathcal{S}^2(T^\ast M))\), its value \(g_p\) is an element of \(\mathcal{S}^2(T^\ast_pM)\). Now, through an argument identical to the one verified after [Manifolds] §Differential Forms, ⁋Definition 1, we see that \(\mathcal{S}^2(T^\ast_pM)\cong(\mathcal{S}^2(T_pM))^\ast\), and by [Multilinear Algebra] §Tensor Algebra, ⁋Proposition 7, we can view \(g_p\) as a symmetric multilinear map from \(T_pM\times T_pM\) to \(\mathbb{R}\). Therefore, if we impose a suitable positive-definiteness condition on \(g_p\), we can regard it as an inner product defined on \(T_pM\). ([Linear Algebra] §Inner Product Spaces, ⁋Definition 1)

Definition 1 A Riemannian metric on a manifold \(M\) means a smooth section \(g\in\Gamma(\mathcal{S}^2(T^\ast M))\) that is positive-definite in the following sense:

(Positive-definiteness) For every \(p\in M\), \(g_p(v,v)>0\) holds for all non-zero \(v\in T_pM\).

A smooth section \(g \in \Gamma(\mathcal{S}^2(T^\ast M))\) obtained by weakening the positive-definiteness condition in the definition above to non-degeneracy is called a pseudo-Riemannian metric. In this case, \(g_p\) is no longer an inner product, but it defines a non-degenerate symmetric bilinear form on \(T_pM\).

As seen above, if \(g\) is a Riemannian metric, then at any point \(p\), \(g_p(-,-)\) defines an inner product on \(T_pM\), which we denote simply by \(\langle -,-\rangle_g\). Also, a manifold equipped with a Riemannian metric \(g\) is written as a pair \((M,g)\) and called a Riemannian manifold.

In particular, if around a point \(p\) we choose a coordinate system \((U,(x^i))\), then \(g\) can be expressed in the form

\[g=\sum_{i,j=1}^ng_{ij}\dd{x}^i\otimes \dd{x}^j\]

and here, \(g\) being a Riemannian metric is equivalent to the \(n\times n\) matrix \((g_{ij})\) being symmetric and positive-definite.

Suppose that on a vector space \(V\), two inner products \(g\) and \(g'\) are given. Then \(g+g'\), defined by

\[(g+g')(v,w)=g(v,w)+g'(v,w)\]

is also an inner product. Also, if \(g\) is an inner product, then for any \(\alpha>0\), multiplying by the constant \(\alpha\) yields \(\alpha g\), which is also an inner product. Now, since an inner product exists on Euclidean space, for any manifold \(M\) we can well-define an inner product on each coordinate chart \((U,\varphi)\), and by summing them all via a partition of unity, we can construct a smooth section of \(\mathcal{S}^2(T^\ast M)\). By the preceding observation, this section is a Riemannian metric. That is, every manifold can always be given a Riemannian metric.

Musical isomorphism

From an algebraic perspective, one of the best consequences of a non-degenerate pairing is that it induces an isomorphism between \(V\) and its dual space \(V^\ast\). ([Linear Algebra] §Bilinear Forms, §§Non-degenerate Bilinear Forms) Similarly, given a Riemannian metric \(g\), \(g\) induces via

\[\tilde{g}:TM\rightarrow T^\ast M;\qquad(p,v)\mapsto (p,\langle v,-\rangle)\tag{1}\]

an isomorphism between the two bundles \(TM\) and \(T^\ast M\). Through this, given any vector field \(X\), we obtain a smooth section of \(T^\ast M\), \(\tilde{g}(X)\).

To examine this in more detail, let us fix a coordinate chart \((x^i)\). Then for any two vector fields

\[X=\sum_{i=1}^n X^i\frac{\partial}{\partial x^i},\quad Y=\sum_{i=1}^n Y^i\frac{\partial}{\partial x^i}\]

we have

\[\tilde{g}(X)(Y)=\sum_{i,j=1}^ng_{ij}\dd{x}^i(X)\dd{x}^j(Y)=\sum_{i,j=1}^ng_{ij}X^iY^j\]

Now, plugging into \(Y\) the vectors \(\partial/\partial x^j\), we see that \(\tilde{g}(X)\) is given by

\[\tilde{g}(X)=\sum_{i,j=1}^n g_{ij}X^i\dd{x}^j\]

Often, \(\sum_{i=1}^ng_{ij}X^i\) is abbreviated as \(X_j\); then the expression above becomes \(\tilde{g}(X)=\sum_{j=1}^nX_j \dd{x}^j\), so it looks as if the index of \(X^i\) has been lowered. For this reason, with a slight play on notation, we denote the covector field \(\tilde{g}(X)\) by \(X^\flat\).

Of course, since (1) is an isomorphism, given any covector field \(\omega\), one can also obtain the corresponding vector field. This vector field is (naturally) denoted by \(\omega^\sharp\), and together these two are called the musical isomorphism. Of course, they are inverses of each other.

Length of Curves

Meanwhile, a Riemannian metric finally allows us to do geometry on manifolds, such as measuring distances and angles. Recall that once an inner product is defined on any vector space \(V\), via \(\lVert v\rVert:=\sqrt{\langle v,v\rangle}\) we could endow \(V\) with a metric.

Definition 2 Let \((M,g)\) be a Riemannian manifold, and let \(\gamma:[a,b]\rightarrow M\) be a piecewise \(C^\infty\) curve defined on it. Then, for \(\gamma\), the length \(\length(\gamma)\) is defined by

\[\length(\gamma)=\int_a^b\lVert\dot{\gamma}(t)\rVert_g\dd{t}\]

The length of a curve defined in this way does not depend on the parametrization. Meanwhile, if \(M\) is connected, we can make \(M\) into a metric space via the definition above. To this end, it suffices to define

\[d_g(p,q)=\inf_{\gamma\text{ connecting }p,q}\length(\gamma)\]

Normal bundle

Finally, we can define the concept of a normal bundle. Suppose a Riemannian manifold \(M\) is given, and consider a submanifold \(S\). Then by restricting \(g\) to \(S\), we obtain on \(S\) a Riemannian metric \(\iota^\ast g\). Through the relation induced by \(\iota\),

\[\dd{\iota}(T_pS)\subseteq T_pM\]

we can view \(T_pS\) as a subspace of \(T_pM\), and therefore \(T_pS\) is a direct summand of \(T_pM\). In general, there is no canonical way to provide a complementary subspace of \(T_pS\), but when \(T_pM\) is an inner product space as it is now, we can define it to be \((T_pS)^\perp\). To each point \(p\), attaching such a subspace \((T_pS)^\perp\) yields a vector bundle over \(\iota(S)\), which we call the normal bundle of \(S\) and write as \(NS\).

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References

[Lee] John M. Lee. Introduction to Riemannian Manifolds, Graduate texts in mathematics, Springer, 2019

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