Riemannian Metric

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Riemannian Metric

We defined the exterior algebra bundle

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

using exterior algebra in §Differential Forms, and defined a smooth section of this bundle as a differential form. The same construction works with the symmetric algebra; unlike the exterior case, the case \(k=2\) is of particular interest. This is because the elements of \(\mathcal{S}^2(T^\ast M)\) that arise when \(k=2\) define symmetric bilinear forms on \(TM\).

Fix a point \(p\in M\). Then \(g_p\) is an element of \(\mathcal{S}^2(T^\ast_pM)\). By the same argument we verified after §Differential Forms, ⁋Definition 1, we see that \(\mathcal{S}^2(T^\ast_pM)\cong(\mathcal{S}^2(T_pM))^\ast\), and by §Tensor Algebras, ⁋Proposition 11, we may regard \(g_p\) as a symmetric multilinear map from \(T_pM\times T_pM\) to \(\mathbb{R}\). Hence, provided we impose an appropriate non-degeneracy condition on \(g_p\), we may view it as an inner product on \(T_pM\). (§Inner Product Spaces, ⁋Definition 1)

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

(Positive-definiteness) For any \(p\in M\), we have \(g_p(v,v)>0\) for all nonzero \(v\in T_pM\).

If we weaken the positive-definiteness condition above to non-degeneracy, the resulting smooth section \(g \in \Gamma(\mathcal{S}^2(T^\ast M))\) 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 observed above, if \(g\) is a Riemannian metric, then at each point \(p\) the map \(g_p(-,-)\) defines an inner product on \(T_pM\); we denote it simply by \(\langle -,-\rangle_g\).

In particular, if we take a coordinate system \((U,(x^i))\) in a neighborhood of a point \(p\), then we can express \(g\) in the form

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

and \(g\) is a Riemannian metric if and only if the \(n\times n\) matrix \((g_{ij})\) is symmetric and positive-definite.

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

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

is also an inner product. Moreover, if \(g\) is an inner product and \(\alpha\) is a nonzero constant, then \(\alpha g\) is again an inner product. Since Euclidean space carries an inner product, we can define an inner product on each coordinate chart \((U,\varphi)\) of any manifold \(M\), and by adding these together via a partition of unity we obtain a function on \(M\). By the preceding observation, this function is a Riemannian metric. Thus every manifold admits a Riemannian metric.

Musical Isomorphism

From an algebraic standpoint, one of the most useful consequences of a non-degenerate pairing is that it induces an isomorphism between \(V\) and its dual space \(V^\ast\). (§Bilinear Forms, §§Non-degenerate Bilinear Forms) Likewise, if a Riemannian metric \(g\) is given, then \(g\) induces an isomorphism between the bundles \(TM\) and \(T^\ast M\) via the formula

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

Consequently, given any vector field \(X\), we obtain a smooth section \(\tilde{g}(X)\) of \(T^\ast M\).

To examine this in more detail, 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

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

Substituting \(\partial/\partial x^j\) for \(Y\), we see that \(\tilde{g}(X)\) is given by

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

We sometimes abbreviate \(\sum_{i=1}^ng_{ij}X^i\) as \(X_j\); the above formula then becomes \(\tilde{g}(X)=\sum_{j=1}^nX_j dx^j\), so the index of \(X^i\) appears to have been lowered. For this reason, with a slight notational convention the covector field \(\tilde{g}(X)\) is denoted by \(X^\flat\).

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

Length of a Curve

A Riemannian metric finally allows us to do geometry on a manifold: measuring distances, angles, and so forth. Recall that once an inner product is defined on a vector space \(V\), we can endow \(V\) with a norm via \(\lVert v\rVert:=\sqrt{\langle v,v\rangle}\).

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

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

The length of a curve defined in this way is independent of the parametrization. Moreover, via the above definition we can make \(M\) into a metric space by setting

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

Normal Bundle

Finally, we can define the notion of a normal bundle. Let \(M\) be a Riemannian manifold and let \(S\) be a submanifold. Restricting \(g\) to \(S\) yields a Riemannian metric \(\iota^\ast g\) on \(S\). The inclusion \(\iota\) induces

\[d\iota(T_pS)\subseteq T_pM\]

so we may regard \(T_pS\) as a subspace of \(T_pM\), and hence \(T_pS\) is a direct summand of \(T_pM\). In general there is no canonical way to choose a complementary subspace of \(T_pS\), but because \(T_pM\) is now an inner product space we can define it as \((T_pS)^\perp\). The vector bundle over \(\iota(S)\) with the fibers \((T_pS)^\perp\) attached at each point \(p\) is called the normal bundle of \(S\) and is denoted by \(NS\).

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References

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

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