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The following notes are intended to supplement the material in the text and illustrate how concepts such as the first fundamental form of a surface and local isometries between smooth surfaces can be generalized.

Linear and bilinear functionals If \(V\) is a finite dimensional vector space, a linear functional on \(V\) is a linear transformation from \(V\) into \(\mathbb{R}\). The collection of all linear functionals is denoted \(V^*\) and is itself a vector space and its dimension is the same as \(V\)'s. A bilinear functional on \(V\) is a function \(\phi:V \times V \to \mathbb{R}\) such that \(\mathbf{u} \mapsto \phi(\mathbf{v},\mathbf{u})\) and \(\mathbf{v} \mapsto \phi(\mathbf{v},\mathbf{u})\) are both linear functionals. We say that \(\phi\) is symmetric if \(\phi(\mathbf{u},\mathbf{v}) = \phi(\mathbf{v},\mathbf{u})\). If \(\phi,\psi \in V^*\), then we define \(\phi \odot \psi : V \times V \to \mathbb{R}\) by \[ \phi \odot \psi(\mathbf{u},\mathbf{v}) := \frac{1}{2} (\phi(\mathbf{u}) \psi(\mathbf{v})+ \phi(\mathbf{v}) \psi(\mathbf{u})) \] This is always a symmetric bilinear functional.

1-forms Suppose now that \(M \subseteq \mathbb{R}^n\) is a smooth \(k\)-dimensional manifold and \(\sigma\) is a regular chart for \(M\). If \(i \leq d\), then \(dx_i\) is the function which sends \(\mathbf{p} \in M\) to the linear functional defined by \[ (dx_i)_{\mathbf{p}} (\sum_{j=1}^k a_j \sigma_{x_j}) := a_i \] A 1-form on \(M\) is a function \(\omega\) defined on \(M\) such that for each \(\mathbf{p} \in M\), \(\omega_{\mathbf{p}} \in T_{\mathbf{p}} M^*\) and for each chart \(\sigma\), \(\omega = \sum_{i=1}^k f_i dx_i\) where \(f_i\) is a smooth function on the range of \(\sigma\).

Bilinear forms Let \(M\) and \(\sigma\) be as above. If \(i,j \leq k\), then \(dx_i dx_j\) is the function defined on the range of \(\sigma\) such that \((dx_i dx_j)_{\mathbf{p}} := (dx_i)_{\mathbf{p}} \odot (dx_j)_{\mathbf{p}}\). A symmetric bilinear form is a function \(\langle\cdot,\cdot\rangle\) such that for each \(\mathbf{p} \in M\), \(\langle\cdot,\cdot\rangle_{\mathbf{p}}\) is a symmetric bilinear functional on \(T_{\mathbf{p}} M\) and such that, for each chart, there are smooth functions \(f_{i,j}\) such that \(\langle\cdot,\cdot\rangle = \sum_{i,j=1}^k f_{i,j}\ dx_i dx_j\). A symmetric bilinear form \(\langle\cdot,\cdot\rangle\) is nondegenerate if for each \(\mathbf{v} \in T M\), \(\langle\mathbf{v},\mathbf{v} \rangle > 0\).

Riemannian metric Suppose that \(M \subseteq \mathbb{R}^n\) is a smooth \(k\)-dimensional manifold. A Riemannian metric is a symmetric, nondegenerate bilinear form on \(M\). A smooth manifold equipped with a Riemannian metric is called a Riemannian manifold. As noted above in our discussion of bilinear functions, a Riemannian metric \(\langle \cdot,\cdot \rangle\) defines a norm \(\| \cdot \|_{\mathbf{p}}\) on \(T_{\mathbf{p}} M\) for each \(\mathbf{p} \in M\) by \(\| \mathbf{v} \|_{\mathbf{p}} := \sqrt{\langle \mathbf{v},\mathbf{v} \rangle_{\mathbf{p}}}\). Moreover this norm "remembers" the bilinear function \(\langle \cdot,\cdot \rangle_{\mathbf{p}}\): \[ \langle \mathbf{u},\mathbf{v} \rangle_\mathbf{p} = \frac{1}{2} \big( \|\mathbf{u} - \mathbf{v}\|_{\mathbf{p}}^2 - \|\mathbf{u}\|_{\mathbf{p}}^2 - \|\mathbf{v}\|_{\mathbf{p}}^2 \big) \]

The first fundamental form If \(M \subseteq \mathbb{R}^n\) is a smooth manifold, there is natural Riemannian metric which \(M\) inherits from \(\mathbb{R}^n\): \[ \langle \mathbf{u}_{\mathbf{p}},\mathbf{v}_{\mathbf{p}}\rangle_{\mathbf{p}} = \mathbf{u} \cdot \mathbf{v} \] This is called the first fundamental form of \(M\) and is the default Riemannian metric on \(M\) unless another is specified. If \(S \subseteq \mathbb{R}^2\) is a smooth surface with chart \(\sigma(u,v)\), then the first fundamental form can be expressed in local coordinates as \( E\ du^2 + 2F\ du\ dv + G\ dv^2\) where \[ E := \sigma_u \cdot \sigma_u = \|\sigma_u\|^2 \qquad \qquad F := \sigma_u \cdot \sigma_v \qquad \qquad G := \sigma_v \cdot \sigma_v = \|\sigma_v\|^2\]

The hyperbolic plane The advantage of working with Riemannian metrics abstractly is that they allow one to easily specify a geometry without specifying an embedding into \(\mathbb{R}^n\). Let \(\mathbb{H}\) be the set of all \((x,y) \in \mathbb{R}^2\) equipped with the Riemannian metric \(\frac{1}{y^2} dx^2 + \frac{1}{y^2} dy^2\). This is the hyperbolic plane and is explored in Homework 6.

Lengths of curves If \(M\) is a Riemannian manifold with a Riemannian metric \(\langle \cdot , \cdot \rangle^M\) and \(\gamma:[a,b] \to M\) is a smooth curve, then the length of \(\gamma\) with respect to the metric is \[ \ell^M(\gamma) := \int_a^b \| \dot \gamma(t) \|_{\gamma(t)}^M \ dt \] If \(M\) is clear from the context, we'll write \(\ell(\gamma)\) for \(\ell^M(\gamma)\).

Pulling back metrics If \(M\) and \(N\) are smooth manifolds, \(f:M \to N\) is a smooth function, and \(\langle \cdot , \cdot \rangle^N\) is a Riemannian metric on \(N\), then \(f\) allows us to define a Riemannian metric \(f^* \langle \cdot,\cdot \rangle^N\) on \(M\) by \[ f^* \langle \mathbf{u},\mathbf{v} \rangle^N :=\langle Df \mathbf{u},Df \mathbf{v} \rangle^N \] where \(\mathbf{u},\mathbf{v} \in T M\). Note that \(f^* \langle \cdot,\cdot \rangle^N\) should be regarded as a single symbol - we are applying \(f^*\) to the metric itself, not composing the metric with \(f^*\). Similarly we define \(f^* \| \cdot \|^N\) by \[ f^* \| \mathbf{v} \|^N := \|Df \mathbf{v} \|^N \]

Local isometries If \(M\) and \(N\) are Riemannian manifolds and \(f:M \to N\) is a smooth function, we say that \(f\) is a (local) isometry if \(f\) is a (local) diffeomorphism and whenever \(\gamma:[a,b] \to M\) is a smooth curve, \(\ell^M (\gamma) = \ell^N(f \circ \gamma)\). That is, \(f\) preserves the lengths of curves.

Theorem Suppose that \(M\) and \(N\) are Riemannian manifolds and \(f:M \to N\) is a local diffeomorphism. The following are equivalent:

1. \(f\) is a local isometry.
2. \(f^* \| \cdot \|^N = \| \cdot \|^M\).
3. \(f^*\langle \cdot,\cdot\rangle^N = \langle \cdot,\cdot \rangle^M\).

Proof To see (1) implies (2), suppose that \(f\) is a local isometry. Let \(\mathbf{v} \in TM\) be arbitrary and let \(\alpha:(-\epsilon,\epsilon) \to M\) be such that \(\dot \alpha(0)= \mathbf{v}\). If \(\beta = f \circ \alpha\), then \(\dot \beta(0) = Df \mathbf{v}\). By (1) we have that any \(a < 0 < b\): \[ \int_a^b \| \dot \alpha \|^M\ dt = \int_a^b \| \dot \beta \|^N \ dt \] This is only possible if \[ \|\mathbf{v}\|^M = \| \dot \alpha(0) \|^M = \|\dot \beta(0) \|^N = \|Df \mathbf{v}\|^N \] Since \(\mathbf{v}\) was arbitrary, \(\|\cdot\|^M = f^* \|\cdot \|^N\).
We have already noted that (2) implies (3) and trivially (3) implies (2). To see that (2) implies (1), let \(\alpha :[a,b] \to M\) be any curve and set \(\beta = f \circ \alpha\). \[ \ell^M(\alpha) = \int_a^b \|\dot \alpha\|^M \ dt = \int_a^b f^* \|\dot \alpha\|^N\ dt \] \[ = \int_a^b \|Df \dot \alpha\|^N \ dt = \int_a^b \| \dot \beta \|^N\ dt = \ell^N (\beta) \] This completes the proof.