The definition of a manifold captures the idea that the coordinate systems are local, and constraints the permitted transformations between these local coordinates. In this sense, a surface is a two-dimensional manifold; space-time is a four-dimensional manifold.

The Metric tensor g allows one to define and compute the *infinitesimal* distance on the manifold:

We recall that the physical significance of this is that if we have a small displacement in spacetime (*dt, dx, dy, dz*) then *ds* is the total distance moved, and also that this quantity is an invariant, i.e all observers in any frame of reference will agree on it.

A metric is positive definite if ds^{2} is always positive, and **Riemannian manifolds** have a metric that is positive definite.

##### Special Relativity

We know that in Einstein's Special Relativity, the **space-time interval** is given by the following specific form, as derived from our article The Minkowski metric

where η_{μν} is the Minkowski **metric tensor**.

We recall also that as explained in Minkowski Space-Time, depending on the values of dt, dx, dy, dz the spacetime separation ds^{2} may be positive, negative or zero, which correspond to timelike, lightlike and spacelike intervals respectively. Therefore, the Minkowski spacetime is NOT a Riemannian manifold.

We call the** signature (p,q,r) ** of the metric tensor g the number (counted with multiplicity) of **positive, negative and zero **components of the metric tensor.

We call a **Lorentzian manifold** a manifold in which the **metric signature is (1, N−1)** for a N dimensional space.

Given the fact that the signature of a diagonal matrix is the number of positive, negative and zero numbers on its main diagonal, the **Minkowski metric has a signature (1,3)** in four-dimensional spacetime, and therefore **Minkowski spacetime is a special case of Lorentzian manifold**.

##### General Relativity

Now, what can we say about the curved spacetime in General Relativity?

In General Relativity, as derived from our article Generalisation of the metric tensor in pseudo-Riemannian manifold the metric is:

In this context also **ds ^{2} could be positive, null or negative**, and the earlier discussed notions of time-like, light-like and space-like intervals in special relativity can similarly be used to classify one-dimensional curves through curved spacetime. A time-like curve can be understood as one where the interval between any two infinitesimally close events on the curve is time-like, and likewise for light-like and space-like curves. Technically the three types of curves are usually defined in terms of whether the tangent vector at each point on the curve is time-like, light-like or space-like.

As this metric could be negative but of the form (-,-,+,+) so with a signature (2,2) and not (1,3) it is NOT considered as a Lorentzian metric and as it is NOT positive definite (ds^{2} >0), it can NOT be a Riemaniann metric tensor neither.

The metric which determines the curved spacetime of General Relativy is a **pseudo-Riemannian metric**, attached to a so called **pseudo-Riemannian manifold**.

**Remark**: a Lorentzian manifold in which the signature of the metric is (1, n−1) (equivalently, (n−1, 1) is a special case of a pseudo-Riemannian manifold but obviously the inverse is not true.

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