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 [Differentiable Manifolds] §Differential Forms, and defined a smooth section of this bundle as a differential form. We can do something similar using the symmetric algebra, and unlike the exterior algebra, the case \(k=2\) is of interest. This is because the elements of \(\mathcal{S}^2(T^\ast M)\) arising when \(k=2\) define symmetric bilinear forms on \(TM\).

Fix a point \(p\in M\). Then \(g_p\) is an element of \(\mathcal{S}^2(T^\ast_pM)\). Now, by the same argument we checked after [Differentiable Manifolds] §Differential Forms, ⁋Definition 1, we can see that \(\mathcal{S}^2(T^\ast_pM)\cong(\mathcal{S}^2(T_pM))^\ast\), and by [Multilinear Algebra] §Tensor Algebras, ⁋Proposition 11, we can think of \(g_p\) as a symmetric multilinear map from \(T_pM\times T_pM\) to \(\mathbb{R}\). Therefore, if we only impose an appropriate non-degeneracy 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 \(M\) 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 \(p\in M\), \(g_p(v,v)>0\) holds for all nonzero \(v\in T_pM\).

If we weaken the positive-definiteness condition in the above definition to non-degeneracy, the 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 we saw above, if \(g\) is a Riemannian metric, then at any point \(p\), \(g_p(-,-)\) defines an inner product on \(T_pM\), so 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 in this case, \(g\) being a Riemannian metric is equivalent to the \(n\times n\) matrix \((g_{ij})\) being symmetric and positive-definite.

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

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

is also an inner product. Also, if \(g\) is an inner product, then \(\alpha g\), where \(\alpha\) is a nonzero constant, is also an inner product. Now since Euclidean space has an inner product, we can define an inner product on each coordinate chart \((U,\varphi)\) of any manifold \(M\), and by adding them all together via a partition of unity, we can create a function on \(M\). By the preceding observation, this function becomes a Riemannian metric. That is, any manifold always admits a Riemannian metric.

Musical Isomorphism

From an algebraic point of view, one of the best consequences of a non-degenerate pairing is that this pairing induces an isomorphism between \(V\) and its dual space \(V^\ast\). ([Linear Algebra] §Bilinear Forms, §§Nondegenerate Bilinear Forms) Likewise, if a Riemannian metric \(g\) is given, then \(g\) induces an isomorphism between the two bundles \(TM\) and \(T^\ast M\) through the formula

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

Through this, 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

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

Now if we substitute \(\partial/\partial x^j\) for \(Y\), we can see that \(\tilde{g}(X)\) is given by the formula

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

We sometimes abbreviate \(\sum_{i=1}^ng_{ij}X^i\) as \(X_j\); then the above formula becomes \(\tilde{g}(X)=\sum_{j=1}^nX_j dx^j\), so it looks as if the index of \(X^i\) has been lowered. For this reason, with a slight notational trick, we denote the covector field \(\tilde{g}(X)\) 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 together these are called the musical isomorphism. Of course, these two are inverses of each other.

Length of a Curve

Meanwhile, a Riemannian metric finally allows us to do geometry on a manifold, such as measuring distances and angles. Recall that if an inner product is defined on any vector space \(V\), then we can endow \(V\) with a distance 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 the formula

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

The length of a curve defined in this way does not depend on the parametrization. Meanwhile, through the above definition we can make \(M\) into a metric space. To do this, we simply define

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

Normal Bundle

Finally, we can define the notion of a normal bundle. Let an arbitrary Riemannian manifold \(M\) be given, and consider a submanifold \(S\). Then by restricting \(g\) to \(S\), we obtain a Riemannian metric \(\iota^\ast g\) on \(S\). By the

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

induced by \(\iota\), we can regard \(T_pS\) 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 give a complementary subspace of \(T_pS\), but when \(T_pM\) is an inner product space as it is now, we can define it as \((T_pS)^\perp\). The vector bundle over \(\iota(S)\) with such bundles \((T_pS)^\perp\) 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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