I had many courses in my undergraduate studies, but for one reason or another never got to see some riemannian geometry. Which is really a shame, because I always wanted to learn some general relativity. So, here I am, studying riemann geometry for the first time. Since there’s lots of things in it, what follows is really just a summary, primarily for my own understanding.
Let be a smooth manifold of dimension , his tangent bundle, the cotangent bundle, and let denote the space of smooth sections of (i.e. the smooth vector fields on ). Analogously, will denote the smooth sections of the tensor bundle – that is, the smooth tensor fields . Tensors fields are multilinear with respect to . For example, , while are the -forms. We also impose .
1. Riemannian metrics
A Riemannian metric on is a -tensor field (in ) – given in local coordinates by – which is:
- symmetric: for every ;
- positive definite: for every .
It induces an inner product on every tangent space. It also allows to turn a vector field into a 1-form and viceversa, namely (in components)
with the obvious extensions to tensors . Basically . One can define and extend the trace operator as well: it’s the contraction with respect to a pair of covariant-contravariant indices. But allows to define it for pairs of covariant indices as well: if then one defines for example with respect to the first index
and one can extend to operators of arbitrary rank (obviously the indices w.r.t. which one takes contraction have to be specified).
Of course one is interested in the general properties of riemannian manifolds, so there’s an obvious notion of equivalence given by the definition of isometry: a diffeomorphism is an isometry between spaces , if
If is oriented, then the riemannian metric induces the riemannian volume element
unique up to scalar multiples. There’s nothing special about the coordinates, so if is any local frame and the dual frame, one can use , and the volume element is again
This choice has the effect that when the ‘s are an oriented orthonormal basis for .
On an orientable riemannian manifold (with or without boundary) one can rebuild the usual machinery for the Green-Gauss formulas. Define by
and divergence operator by
where is the exterior derivative and the interior multiplication by , i.e. . Then thanks to Stoke’s theorem one can prove
where is the volume element arising from the induced metric on , and is the outward normal to . It’s defined by . Then, since
and defining the laplacian , Green formula follows
With this one has the usual property that the eigenvalues of are non-positive. Harmonic functions on compact connected manifolds without boundary are constants, by Green formula and positive definiteness of .
There’s another structure that can be produced on a manifold , a connection. A (linear, or affine) connection is an operator
- it is -linear in the 1st variable;
- it’s -linear in the 2nd variable;
- the following product rule holds:
It’s easy to see that only depends on the value and on the values of in an arbitrarily small neighbourhood of (we see here the effect of the different linearity in the two variables). In any given frame , one can compute the Christoffel symbols
and then one can express in coordinates ( , the Christoffel symbols calculated in the same coordinate frame) by
The previous relation actually allows to define a connection for every set of smooth functions . Notice that in the euclidean case of , the Christoffel symbols are identically .
Given a connection one can extend it uniquely to maps
by imposing the following further properties:
- if , then ;
- Product rule holds for tensor products:
- traces/contractions commute with :
With these properties, one then has that a product rule also holds for the natural pairing of forms and vectors,
and for a general tensor field it is
As an example, in coordinates,
From a linear connection one can define an operator called total covariant derivative as follows:
For example, if then . is called the covariant hessian of , and it holds . From a coordinate point of view, one can write (commonly accepted notation is a very unfortunate one) in the case of a vector field :
As it will be useful in what follows, here’s the coordinate expression for (it holds for any 2-tensor actually):
The importance of a connection is in that it allows one to define geodesics. One might be tempted to define them through the usual length minimization properties, but that’s a global feature, and we rather need a local one. Using a physical analogy, a point along a geodesic experiences no acceleration. But acceleration, unlike velocity, can’t be calculated by taking the limit of a quotient – as the numerator would contain the difference of vectors that live in different tangent spaces, which makes no sense. A connection allows to avoid this problem by defining derivations along curves.
Let be a smooth curve in , and let Then we can define the covariant derivative along at as
The definition is well posed because actually only depends on and the values of along a curve through with tangent , rather than from an entire neighbourhood of (it’s immediate from the coordinate expression for ). If the vector field is defined only on that’s no trouble, since it can always be extended to a vector field on . This definition implies the product rule
where . In local coordinates
Now, acceleration is , thus the condition for to be a geodesic is that
which can be immediately translated into coordinates:
These are the geodesic equations, which are ODEs. By standard ODE theory then, given initial conditions , , one has that a geodesic through with tangent exists, and is unique in its maximal domain of existence.
In general, a vector field in such that is said to be parallel along – so, a geodesic is a curve whose velocity is parallel along the curve itself. The following holds: if , and is a curve, there exists a unique vector field in parallel along to . Thus given a curve one can define the operator of parallel transport from to
which is a linear isomorphism of tangent spaces. With this operator one can actually compute as the limit of a quotient:
4. Riemannian connection
Given a linear connection one can define the following tensor field in ,
called the torsion tensor. It is indeed a tensor field as it’s -linear in both variables. By writing down local coordinates, one sees that is identically 0 iff the Christoffel symbols are symmetric in the lower indices (in a coordinate frame, not necessarily in any frame ), in which case we say is symmetric. All in all, the torsion tensor measures how much $\nabla$ fails to be symmetric.
So far and are separate objects, and as we’ve seen there’s plenty of choices for itself. We boil it down to just one by imposing two conditions:
- is compatible with , i.e.
- is symmetric.
The 1st condition is equivalent to a number of other conditions:
- if for some curve then
- if are parallel along then is constant along ;
- parallel transport operators are all isometries.
For example, the 1st one is equivalent because . As for the 2nd, extending to vector fields and taking the compatibility relation implies
which is exactly the condition in 2., since . The other way round is similar. Now is immediate, and finally by taking the derivative of .
As stated above, compatibility + symmetry completely define a unique connection, the riemannian connection (or Levi-Civita’s):
Theorem 1 (Fundamental lemma of Riemannian geometry) Let be a riemannian manifold. Then there exists a unique connection that is compatible with and symmetric.
Proof: By compatibility
and cyclic permutations, but by symmetry
Thus adding a cyclic permutation and subtracting the other
where as you can notice the r.h.s. doesn’t depend on at all (which implies uniqueness). Now, choosing ,
because , thus
Symmetry is evident, compatibility may be directly verified by substituting into the previous expression for .
As a corollary, by the equivalent condition stated above, geodesics w.r.t. the riemannian connection have constant speed:
In the related field of pseudo-riemannian geometry, where the positive definiteness of is relaxed to non-degeneracy, virtually all results that don’t require to be positive hold as well. In applications to physics (general relativity) one considers 4-dimensional manifolds with metrics of signature . The pseudo-riemannian connection is built the same way, and thus geodesics exhibit the same constant speed property, but now it can be that . In the case one calls the geodesic null-like: this is interpreted as the path a ray of light would take. Speed of light is thus constant when measured by any observer.
Notice also that at this point we are already able to compute (at least numerically, hopefully) the geodesics of a manifold, given : first compute the Christoffel symbols of the Levi-Civita’s connection, then solve the geodesic equations for given initial values.
, where . Just to clarify, is a real-valued multilinear map which takes as arguments vectors and covectors.
It is customary to denote by the inverse matrix of . Its elements are the coefficients of a tensor, .
The result again is a consequence of standard ODE theory, but notice that one has existence along all of the curve because the system is linear in .
One can work out the formula for the Christoffel symbols’ transformation law under a change of coordinates , namely