The Riemannian metric

Recall that a (pseudo) metric tensor is a (pseudo) inner product \({\left\langle v,w\right\rangle }\) on a vector space \({V}\) that can be represented by a symmetric tensor \({g_{ab}}\), and thus can be used to lower and raise indices on tensors. A (pseudo) Riemannian metric (AKA metric) is a (pseudo) metric tensor field on a manifold \({M}\), making \({M}\) a (pseudo) Riemannian manifold. The metric and manifold are also described by the same terms used to characterize the signature, i.e. Lorentzian manifold, Minkowskian metric, etc.

For a (non-pseudo) Riemannian manifold, a metric defines the length (norm) of tangent vectors, and can thus be used to define the length \({L}\) of a curve \({C}\) via parametrization and integration:

\(\displaystyle \begin{aligned}L(C) & \equiv\int\left\Vert \dot{C}(t)\right\Vert \mathrm{d}t\\ & =\int\sqrt{\left\langle \dot{C}(t),\dot{C}(t)\right\rangle }\mathrm{d}t \end{aligned} \)

This also turns any Riemannian manifold into a metric space, with distance function \({d(x,y)}\) defined to be the minimum length curve connecting the two points \({x}\) and \({y}\); this curve is called a Riemannian geodesic (AKA geodesic), and it locally minimizes the distance between any of its points. This distance is only locally minimal since e.g. the curve may eventually self-intersect as the equator on a sphere does. It can be shown that for any tangent vector \({v}\) on a Riemannian manifold there is a unique geodesic \({C_{v}(d)}\) parametrized by distance whose tangent is \({v}\); one can then define the exponential map by \({\mathrm{exp}(v)\equiv C_{v}(1)}\).

With a metric, our intuitive picture of a manifold loses its “stretchiness” via the introduction of length and angles; but having only intrinsically defined properties, the manifold can still be e.g. rolled up like a piece of paper if imagined as flat and embedded in a larger space.

Recalling the component expression for the unit \({n}\)-vector, the volume of an region \({U\in M^{n}}\) corresponding to \({R\in\mathbb{R}^{n}}\) in the coordinates \({x^{\mu}}\) is

\(\displaystyle V(U)=\int_{R}\sqrt{\left|\mathrm{det}(g)\right|}\mathrm{d}x^{1}\ldots\mathrm{d}x^{n}, \)

where \({\mathrm{det}(g)}\) is the determinant of the metric tensor as a matrix in the coordinate frame \({\partial/\partial x^{\mu}}\). In the context of a pseudo-Riemannian manifold, the integrand \({\mathrm{d}V\equiv\sqrt{\left|\mathrm{det}(g)\right|}\mathrm{d}x^{1}\ldots\mathrm{d}x^{n}}\) is called the (pseudo-Riemannian) volume element, or when written as a form \({\mathrm{d}V\equiv\sqrt{\left|\mathrm{det}(g)\right|}\mathrm{d}x^{1}\wedge\cdots\wedge\mathrm{d}x^{n}}\) it is called the volume form. In physical applications \({\mathrm{d}V}\) usually denotes the volume pseudo-form, which gives a positive value regardless of orientation and which can be integrated over non-orientable regions. Note that if the coordinate frame is orthonormal then \({\left|\mathrm{det}(g)\right|=1}\); thus these definitions are consistent with those previously given. Also note that these definitions are all relative to a specified metric; if one defines a volume form on a manifold without defining a metric, the metric (and connection) is not uniquely determined.

We can use the inner product to define an orthonormal frame on \({M}\). In four dimensions an orthonormal frame is also called a tetrad (AKA vierbein). Any frame on a manifold can be defined to be an orthonormal frame, which is equivalent to defining the metric (which in the orthonormal frame is \({g_{\mu\nu}=\eta_{\mu\nu}}\)). An orthonormal holonomic frame exists on a region of \({M}\) if and only if that region is flat. Thus in general, given a set of coordinates on \({M}\), we have to choose between using either a non-coordinate orthonormal frame or a non-orthonormal coordinate frame.

The Hopf-Rinow theorem says that a connected Riemannian manifold \({M}\) is complete as a metric space (or equivalently, all closed and bounded subsets are compact) if and only if it is geodesically complete, meaning that the exponential map is defined for all vectors at some \({p\in M}\). If \({M}\) is geodesically complete at \({p}\), then it is at all points on the manifold, so this property can also be used to state the theorem. This theorem is not valid for pseudo-Riemannian manifolds; any (pseudo) Riemannian manifold that is geodesically complete is called a geodesic manifold.

As noted previously, a Riemannian metric can be defined on any differentiable manifold. In general, however, not every manifold admits a pseudo-Riemannian metric, and in particular not every 4-manifold admits a Minkowski metric; but 4-manifolds that are noncompact, parallelizable, or compact, connected and of Euler characteristic 0 all do.

In the same way that differentiable manifolds are equivalent if they are related by a diffeomorphism, Riemannian manifolds are equivalent if they are related by an isometry, a diffeomorphism \({\Phi\colon M\rightarrow N}\) that preserves the metric, i.e. \({\forall v,w\in TM}\), \({\left\langle v,w\right\rangle \left|_{p}\right.=\left\langle \mathrm{d}\Phi_{p}(v),\mathrm{d}\Phi_{p}(w)\right\rangle \left|_{\Phi(p)}\right.}\). Also like diffeomorphisms, the isometries of a manifold form a group; for example, the group of isometries of Minkowski space is the Poincaré group. A vector field whose one-parameter diffeomorphisms are isometries is called a Killing field, also called a Killing vector since it can be shown ([16] pp. 188-189) that a Killing field is determined by a vector at a single point along with its covariant derivatives. A Killing field thus satisfies \({L_{v}g_{ab}=0}\), which for a Levi-Civita connection (see next section) is equivalent to \({\nabla_{a}v_{b}+\nabla_{b}v_{a}=0}\), called the Killing equation (AKA Killing condition).

We can then consider isometric immersions and embeddings, and ask whether every Riemannian manifold can be embedded in some \({\mathbb{R}^{n}}\). The Nash embedding theorem provides an affirmative answer, and it can also be shown that every pseudo-Riemannian manifold can be isometrically embedded in some \({\mathbb{R}^{n}}\) with some signature while maintaining arbitrary differentiability of the metric.

Δ A potential source of confusion is that the terms isometry and isometric are also used to refer to maps between metric spaces which preserve the distance function. For example, \({S^{2}}\) can be isometrically embedded in \({\mathbb{R}^{3}}\) with regard to the Riemannian metric, but not with regard to the distance function.

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