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.