Notions of curvature in geometry

Here will be explained the two notions of curvature involved in geometry, and needed in the description of our space-time by General Relativity :

The extrinsic curvature

When following a curve, its direction (tangent vector) can vary. The angle of deviation of this direction for an arc of a smooth curve S near a given point P, is usually proportional to its length, either exactly (in the case of an arc of circle), or approximately (the approximation getting better as the arc is smaller). The extrinsic curvature of S at P is defined as the ratio of this deviation (proportionality coefficient) :

Limit (for small arcs of S containing P), of (Angle of deviation (in radians) / Arc length).

The curvature of a circle in an Euclidean plane, is the inverse of its radius.
In a sphere, the "straight lines", curves with zero extrinsic curvature, are the "big circles" with the same center as the sphere. On the Earth approximated as a sphere, the equator and meridians are straight, but parallels (circles of latitude) are curved.
We have an intuitive understanding of the curvature of a curve in a plane, by taking the case of a road on a horizontal ground, that a car is following: the curvature of the road near each point is "measured" by the orientation of the wheel that is needed to follow it, and also by the sensation of lateral push when following it at a constant speed.

For a subspace S of E, the extrinsic curvature of S in E is the measure of how a curve "straight inside S" is curved inside E. When S has dimension > 1, this curvature is generally not a single quantity but a multidimensional object, as it gives a direction orthogonal to (the tangent subspace of) S as a (quadratic) function of the chosen direction of curve inside S (precisely, if n = dim S and m = dim E then the extrinsic curvature has n(n+1)(mn)/2 coordinates). But it may be summed up as a single quantity when it is the same in all directions.

The Gaussian (intrinsic) curvature

The Gaussian curvature is the intrinsic curvature for the particular case of a surface (2-dimensional space), where it is a scalar field.

The Gaussian curvature can be thought of as a "density of angles", as follows.

In any surface (2-dimensional space, only approximately Euclidean in small scales), the sum of angles (in radians) of any triangle, may differ from π. For example, on Earth, a triangle made of a part of the equator and of 2 meridians, has two right angles, and the remaining angle (at the pole) can have any value.
More generally, the parallel transport along any closed curve induces a rotation with angle α, which in the case of a triangle coincides with this difference of angles:

α = (∑ angles) − π.

In the case of a (2-dimensional) sphere with radius r inside a 3-dimensional Euclidean space, this angle α is proportional to the area A of the surrounded region, according to the formula

 α = r-2 A

The Gaussian curvature of this sphere is the proportionality coefficient R = r-2. It equals the square of the extrinsic curvature (r-1).

Spheres are a particular case of curved surface, having a constant Gaussian curvature. In the more general case, the Gaussian curvature of a surface is a (non-constant) field, and the rotation angle α induced by a parallel transport along a closed curve, is the integral of this field over the surrounded surface.

The Riemann curvature

The Riemann curvature is the general case of intrinsic curvature, for spaces with any dimension (higher than 2). It is a field that is not scalar but multidimensional (described by several components when a coordinates system is given). For a 3-dimensional space, the Riemann curvature around each point has dimension 6. For a 4-dimensional space (such as our space-time), it has dimension 20.

There are different ways to describe this field. The standard, "mathematical" way, is by properties of the covariant derivative operator, that is the operator describing the variations of any vector field at the neighborhood of each point.
Generally, metaphorically describing a field as something to be "measured" at every point, the variations of a field of any kind of objects (vectors or something else), are described by a linear operator from the space of "speed vectors" of a measuring device passing through the given point, whose value (inside the vector space where the field takes values) is the speed of variation of the measured value of the field.

The covariant derivative of a vector field at every point, is the "most faithful" picture of the variation of the field around this point, in the sense that it only fits with the partial derivatives of this field if we choose a coordinates system which is specially "least distorted" near this point.
The properties of the covariant derivative, differs from those of the partial derivatives in any fixed coordinates system, because (if the curvature is nonzero), a coordinates system cannot be "least distorted" in all the neighborhood of a point, so that in the neighborhood we cannot keep a fixed coordinates system that makes the covariant derivative directly given by partial derivatives.
In other words, the variations measured by partial derivatives, are those coming from the affine structure of the space that is given by the chosen coordinates system; but that affine structure is that of a flat geometry, with no curvature, that is not the one we want to study.

But now we are going to give an intuitive description of the curvature, by extending to higher dimensions the above idea of parallel transport along a curve.

For every little loop near a point, let us imagine it as a solid loop that we cut at a point and then move into a flat geometrical space. Both ends of the cut, that initially coincided, are now apart from each other.
To make them fit again as they initially did, we can cut the loop at another point, thus making 2 pieces, and then move one piece relatively to the other.
This movement that must be applied to one piece relatively to the other, to make both ends fit (but the ends of the new cutting point go apart), is a rotation (generally, a small Euclidean move, that rotates more than translates, in the sense that, if a rotation, its center is in or near the considered region).
In first approximation (for a small loop), this rotation (its coordinates: direction, small angle(s)...), does not depend on the point where this loop was cut.

Now as loops we shall take small parallelograms. The effect of parallel transport on any loop can be deduced from that on small parallelograms, by taking a surface bordered by this loop and dividing this surface into a large number of small parallelograms (or triangles, which count as half a parallelogram): the global rotation by parallel transport around the loop is obtained by somehow summing up (integrating) all the small rotations coming from parallel transport around each parallelogram.

The operation which to each pair of (small) vectors, associates the (small) rotation produced by the parallel transport along a small parallelogram whose direction is given by these vectors, is a bilinear, antisymmetric function of these vectors. 

The small rotations (values of this function), can be themselves identified as bilinear antisymmetric forms, described by antisymmetric matrices.

To describe the Riemann curvature by coordinates, let us take an approximate "Cartesian coordinates system" of a small neighborhood (in the case of cosmology, it is "small" compared to the universe, but big compared to the galaxy ;-). In fact, we shall use these names of coordinates as the labels of pairwise orthogonal axis in that neighborhood. Then, each pair of axis defines a small parallelogram in this neighborhood.
The space of bilinear antisymmetric forms of an n-dimensional space, has dimension n(n-1)/2, corresponding to the pairs of coordinates.
In the case n=4, such as our space-time with coordinates (x,y,z,t), we have 6 pairs : (x,y), (x,z), (y,z), (x,t), (y,t), (z,t).

Putting all together, the curvature can be described by a tensor with 4 indices, forming two antisymmetric pairs. In coordinates, each component of the Riemann curvature must be labelled by 2 pairs of coordinates :

The Riemann curvature is antisymmetric with respect to the exchange of both directions inside each pair, but it is symmetric with respect to the exchange of both pairs.

So, the Riemann curvature around each point of a 4-dimensional space, in an approximately Cartesian coordinate system around that point, is described by a symmetric 6×6 matrix where each line and each column is labelled by a pair of coordinates.

A special case of curved geometrical spaces, is those with a with constant curvature, equivalently defined by

Now we are ready to explain the expansion of the Universe in General Relativity.

Why dimension 20 rather than 21 ?

The space of symmetric 6×6 matrices has dimension 6×7/2=21.

But the Riemann curvature tensor cannot take any value there, as it is subject to one relation, letting its possible values vary with dimension 20 instead of 21.
This relation says that the "totally antisymmetric" part of that tensor, defined as the sum of components labelled ((x,y),(z,t)), ((y,z),(x,t)) and ((z,x),(y,t)), cancels.

The reason for this, together with that of the symmetry by the exchange of both pairs, can be explained by the way in which the curvature is calculated from the metric structure of the space.

The metric structure is the field which, at each point, gives the local geometric structure (the locally Euclidean or Minkowskian structure, beyond the mere locally affine structure given by the smoothness of the space). The value of this field at each point is thus a symmetric bilinear form between tangent vectors at this point (intuitively, speed vectors for particles passing by), that define the (local) dot product between them.

In a given coordinates system (or equivalently, in a map of the considered space into a flat space - we can make it fit on first approximation around a point, so that the first derivatives of the metric cancel at this point... for a reason a bit technical to explain), the curvature is calculated from the second derivatives of the metric.
The symmetry of the metric, together with the symmetry of the second derivatives in a coordinate system, naturally affects the result of such an expression, that just "cannot produce anything dissymetric" from symmetric entries. The details of this argument, require the understanding of tensor calculus.

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