A (semi-Riemannian) metric on a manifold M is a smooth field g_{ab} on M that is symmetric and invertible; i.e., there exists an (inverse) field g^{bc} on M such that g_{ab}g^{bc} = δ_{a}^{c}.

The inverse field g^{bc} of a metric g_{ab} is symmetric and unique. It is symmetric since

g^{cb} = g^{nb} δ^{n}_{c} = g^{nb}(g_{nm} g^{mc}) = (g_{mn} g^{nb})g^{mc} = δ_{m}^{b} g^{mc} = g^{bc}

(Here we use the symmetry of g^{nm} for the third equality.) It is unique because if g^{′bc} is also an inverse field, then

g^{′bc} = g^{′nc} δ_{n}^{b} = g^{′nc}(g_{nm} g^{mb}) = (g_{mn} g^{′nc}) g^{mb} = δ_{m}^{c} g^{mb} = g^{cb} = g^{bc}

(Here again we use the symmetry of g_{nm} for the third equality; and we use the symmetry of g^{cb} for the final equality.) The inverse field g^{bc} of a metric g_{ab} is smooth. This follows, essentially, because given any invertible square matrix A (over R), the components of the inverse matrix A^{−1} depend smoothly on the components of A.

The requirement that a metric be invertible can be given a second formulation. Indeed, given any field g_{ab} on the manifold M (not necessarily symmetric and not necessarily smooth), the following conditions are equivalent.

(1) There is a tensor field g^{bc} on M such that g_{ab}g^{bc} = δ_{a}^{c}.

(2) ∀ p in M, and all vectors ξ^{a} at p, if g_{ab}ξ^{a} = 0, then ξ^{a} =0.

(When the conditions obtain, we say that g_{ab} is non-degenerate.) To see this, assume first that (1) holds. Then given any vector ξ^{a} at any point p, if g_{ab} ξ^{a} = 0, it follows that ξ^{c} = δ_{a}^{c} ξ^{a} = g^{bc} g_{ab} ξ^{a} = 0. Conversely, suppose that (2) holds. Then at any point p, the map from (M_{p})^{a} to (M_{p})^{b} defined by ξ^{a} → g_{ab} ξ^{a} is an injective linear map. Since (M_{p})^{a} and (M_{p})^{b} have the same dimension, it must be surjective as well. So the map must have an inverse g^{bc} defined by g^{bc}(g_{ab} ξ^{a}) = ξ^{c} or g^{bc} g_{ab} = δ_{a}^{c}.

In the presence of a metric g_{ab}, it is customary to adopt a notation convention for “lowering and raising indices.” Consider first the case of vectors. Given a contravariant vector ξ^{a} at some point, we write g_{ab} ξ^{a} as ξ^{b}; and given a covariant vector η_{b}, we write g^{bc} η_{b} as η^{c}. The notation is evidently consistent in the sense that first lowering and then raising the index of a vector (or vice versa) leaves the vector intact.

One would like to extend this notational convention to tensors with more complex index structure. But now one confronts a problem. Given a tensor α_{c}^{ab} at a point, for example, how should we write g^{mc} α_{c}^{ab}? As α^{mab}? Or as α^{amb}? Or as α^{abm}? In general, these three tensors will not be equal. To get around the problem, we introduce a new convention. In any context where we may want to lower or raise indices, we shall write indices, whether contravariant or covariant, in a particular sequence. So, for example, we shall write α^{ab}_{c} or α^{a}_{c}^{b} or α_{c}^{ab}. (These tensors may be equal – they belong to the same vector space – but they need not be.) Clearly this convention solves our problem. We write g^{mc} α^{ab}c as α^{abm}; g^{mc} α^{a}_{c}^{b} as α^{amb}; and so forth. No ambiguity arises. (And it is still the case that if we first lower an index on a tensor and then raise it (or vice versa), the result is to leave the tensor intact.)

We claimed in the preceding paragraph that the tensors α^{ab}_{c} and α^{a}_{c}^{b} (at some point) need not be equal. Here is an example. Suppose ξ1^{a}, ξ2^{a}, … , ξn^{a} is a basis for the tangent space at a point p. Further suppose α^{abc} = ξi^{a} ξj^{b} ξk^{c} at the point. Then α^{acb} = ξi^{a} ξj^{c} ξk^{b}. Hence, lowering indices, we have α^{ab}_{c} =ξi^{a} ξj^{b} ξk^{c} but α^{a}_{c}^{b} =ξi^{a} ξj_{c} ξi^{b} at p. These two will not be equal unless j = k.

We have reserved special notation for two tensor fields: the index substiution field δ_{b}^{a} and the Riemann curvature field R^{a}_{bcd} (associated with some derivative operator). Our convention will be to write these as δ^{a}_{b} and R^{a}_{bcd} – i.e., with contravariant indices before covariant ones. As it turns out, the order does not matter in the case of the first since δ^{a}_{b} = δ_{b}^{a}. (It does matter with the second.) To verify the equality, it suffices to observe that the two fields have the same action on an arbitrary field α^{b}:

δ_{b}^{a}α^{b} = (g_{bn}g^{am}δ^{n}_{m})α^{b} = g_{bn}g^{an}α^{b} = g_{bn}g^{na}α^{b} = δ^{a}_{b}α^{b}

Now suppose g_{ab} is a metric on the n-dimensional manifold M and p is a point in M. Then there exists an m, with 0 ≤ m ≤ n, and a basis ξ1^{a}, ξ2^{a},…, ξn^{a} for the tangent space at p such that

g_{ab}ξi^{a} ξi^{b} = +1 if 1≤i≤m

g_{ab}ξi^{a}ξi^{b} = −1 if m<i≤n

g_{ab}ξi^{a}ξj^{b} = 0 if i ≠ j

Such a basis is called orthonormal. Orthonormal bases at p are not unique, but all have the same associated number m. We call the pair (m, n − m) the signature of g_{ab} at p. (The existence of orthonormal bases and the invariance of the associated number m are basic facts of linear algebraic life.) A simple continuity argument shows that any connected manifold must have the same signature at each point. We shall henceforth restrict attention to connected manifolds and refer simply to the “signature of g_{ab}”

A metric with signature (n, 0) is said to be positive definite. With signature (0, n), it is said to be negative definite. With any other signature it is said to be indefinite. A Lorentzian metric is a metric with signature (1, n − 1). The mathematics of relativity theory is, to some degree, just a chapter in the theory of four-dimensional manifolds with Lorentzian metrics.

Suppose g_{ab} has signature (m, n − m), and ξ1^{a}, ξ2^{a}, . . . , ξn^{a} is an orthonormal basis at a point. Further, suppose μ^{a} and ν^{a} are vectors there. If

μ^{a} = ∑^{n}_{i=1} μ^{i} ξi^{a} and ν^{a} = ∑^{n}_{i=1} ν^{i} ξi^{a}, then it follows from the linearity of g_{ab} that

g_{ab}μ^{a} ν^{b} = μ1ν1 +…+ μmνm − μ(m+1)ν(m+1) −…−μnνn.

In the special case where the metric is positive definite, this comes to

g_{ab}μ^{a}ν^{b} = μ1ν1 +…+ μnνn

And where it is Lorentzian,

g_{ab} μ^{a}ν^{b} = μ1ν1 − μ2ν2 −…− μnνn

Metrics and derivative operators are not just independent objects, but, in a quite natural sense, a metric determines a unique derivative operator.

Suppose g_{ab} and ∇ are both defined on the manifold M. Further suppose

γ : I → M is a smooth curve on M with tangent field ξ^{a} and λ^{a} is a smooth field on γ. Both ∇ and g_{ab} determine a criterion of “constancy” for λ^{a}. λ^{a} is constant with respect to ∇ if ξ^{n}∇_{n}λ^{a} = 0 and is constant with respect to g_{ab }if g_{ab} λ^{a} λ^{b} is constant along γ – i.e., if ξ^{n} ∇_{n} (g_{ab} λ^{a} λ^{b} = 0. It seems natural to consider pairs g_{ab} and ∇ for which the first condition of constancy implies the second. Let us say that ∇ is compatible with g_{ab} if, for all γ and λ^{a} as above, λ^{a} is constant w.r.t. g_{ab} whenever it is constant with respect to ∇.