In many posts on this blog, such as Geometry on Curved Spaces and Connection and Curvature in Riemannian Geometry, we have discussed the subject of differential geometry, usually in the context of physics. We have discussed what is probably its most famous application to date, as the mathematical framework of general relativity, which in turn is the foundation of modern day astrophysics. We have also seen its other applications to gauge theory in particle physics, and in describing the phase space, whose points corresponds to the “states” (described by the position and momentum of particles) of a physical system in the Hamiltonian formulation of classical mechanics.
In this post, similar to what we have done in Varieties and Schemes Revisited for the subject of algebraic geometry, we take on the objects of study of differential geometry in more technical terms. These objects correspond to our everyday intuition, but we must develop some technical language in order to treat them “rigorously”, and also to be able to generalize them into other interesting objects. As we give the technical definitions, we will also discuss the intuitive inspiration for these definitions.
Just as varieties and schemes are the main objects of study in algebraic geometry (that is until the ideas discussed in Grothendieck’s Relative Point of View were formulated), in differential geometry the main objects of study are the differentiable manifolds. Before we give the technical definition, we first discuss the intuitive idea of a manifold.
A manifold is some kind of space that “locally” looks like Euclidean space . -dimensional Euclidean space is just the line , -dimensional Euclidean space is the plane , and so on. Obviously, Euclidean space itself is a manifold, but we want to look at more interesting examples, i.e. spaces that “locally” look like Euclidean space but “globally” are very different from it.
As an example, consider the surface of the Earth. “Locally”, that is, on small regions, the surface of the Earth appears flat. However, “globally”, we know that it is actually round.
Another way to think about things is that any small region on the surface of the Earth can be put on a flat map (possibly with some distortion of distances). However, there is no flat map that can include every point on the surface of the Earth while continuing to make sense. The best we can do is use several maps with some overlaps between them, transitioning between different maps when we change the regions we are looking at. We want these overlaps and transitions to make sense in some way.
In differential geometry, what we want is to be able to do calculus on these more general manifolds the way we can do calculus on the line, on the plane, and so on. In order to do this, we require that the “transitions” alluded to in the previous paragraph are given by differentiable functions.
Summarizing the above discussion in technical terms, an -dimensional differentiable manifold is a topological space with homeomorphisms from the open subsets covering to , such that the composition is a differentiable function on .
Following the analogy with maps we discussed earlier, the pair is called a chart, and the collection of all these charts that cover the manifold is called an atlas. The map is called a transition map.
Now that we have defined what a manifold technically is, we discuss some related concepts, in particular the objects that “live” on our manifold. Perhaps the most basic of these objects are the functions on the manifold; however, we won’t discuss the functions themselves too much since there are not that many new concepts regarding them.
Instead, we will use one of the most useful concepts when it comes to discussing objects that “live” on manifolds – fiber bundles (see Vector Fields, Vector Bundles, and Fiber Bundles). A fiber bundle is given by a topological space with a projection from to a base space , with the requirement that the space is homeomorphic to the product space , where is the fiber, defined as for any point of . When the fiber is also a vector space, we refer to as a vector bundle. In differential geometry, we require that the relevant maps be also diffeomorphic, i.e. differentiable and bijective.
One of the most important kinds of vector bundles in differential geometry are the tangent bundles, which can be thought of as the collection of all the tangent spaces of a manifold at every point, for all the points of the manifold. We have already made use of these concepts in Geometry on Curved Spaces, and Connection and Curvature in Riemannian Geometry. We needed it, for example, to discuss the notion of parallel transport and the covariant derivative in Riemannian geometry. We will now discuss these concepts more technically.
Let be the ring of real-valued differentiable functions defined in a neighborhood of a point in a differentiable manifold . We define the real tangent space at , written , to be the vector space of -centered -linear derivations, which are -linear maps satisfying Leibniz’s rule . Any such derivation can be written in the following form:
This means that is a basis for the real tangent space at . It might be a little jarring to see “differential operators” serving as a basis for a vector space, but it might perhaps be helpful to think of tangent vectors as giving “how fast” functions on the manifold are changing at a certain point. See the following picture:
The manifold is , and its tangent space at the point is . One of the tangent vectors, , is shown. The parametrized curve is often used to define the tangent vector, although that is not the approach we have given here (it may be found in the references, and is closely related to the definition we have given).
Another concept that we will need is the concept of -forms. A -form on a particular point on the manifold takes a single tangent vector (an element of the tangent space at that particular point) as an input and gives a number as an output. Just as we have the notion of tangent vectors, tangent spaces, and tangent bundles, we also have the “dual” notion of -forms, cotangent spaces, and cotangent bundles, and just as the basis of the tangent vectors are given by , we also have a basis of -forms given by .
Aside from -forms, we also have mathematical objects that take two elements of the tangent space at a point (i.e. two tangent vectors at that point) as an input and gives a number as an output.
An example that we have already discussed in this blog is the metric tensor, which we refer to sometimes as simply the metric (calling it the metric tensor, however, helps prevent confusion as there are many different concepts in mathematics also referred to as a metric). We have been thinking of the metric tensor as expressing the “infinitesimal distance formula” at a certain point on the manifold.
The metric tensor is defined as a symmetric, nondegenerate, bilinear form. “Symmetric” means that we can interchange the two inputs (the tangent vectors) and get the same output. “Nondegenerate” means that, holding one of the inputs fixed and letting the other vary, having an output of zero for all the varying inputs means that the fixed input must be zero. “Bilinear form” means that it is linear in either input – it respects addition of vectors and multiplication by scalars. If we hold one input fixed, it is then a linear transformation of the other input.
In the case of our previous discussions on Riemannian geometry, the output of the metric tensor is a positive real number, expressing the infinitesimal distance. Hence, a metric tensor on a differentiable manifold which always gives a positive real number as an output is called a Riemannian metric. A manifold with a Riemannian metric is of course called a Riemannian manifold.
In general relativity, the spacetime interval, unlike the distance, may not necessarily be positive. More technically, spacetime in general relativity is an example of a pseudo-Riemannian (or semi-Riemannian) manifold, which do not require the metric to be positive (more specifically it is a Lorentzian manifold – we will leave the details of these definitions to the references for now). As we have seen though, many concepts from the study of Riemannian manifolds carry over to the pseudo-Riemannian case.
Another example of these kinds of objects are the differential forms (see Differential Forms). One important example of these objects is the symplectic form in symplectic geometry (see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry), which is used as the mathematical framework of the Hamiltonian formulation of classical mechanics. Just as the metric tensor is related to the “infinitesimal distance”, the symplectic form is related to the “infinitesimal area”.
As an example of the symplectic form, the “phase space” in the Hamiltonian formulation of classical mechanics is made up of points which correspond to a “state” of a system as given by the position and momentum of its particles. For the simple case of one particle constrained to move in a line, the symplectic form (written ) is given by
where is the position and is the momentum, serving as the coordinates of the phase space (by the way, the phase space is itself already the cotangent bundle of the configuration space, the space whose points are the different “configurations” of the system, which we can think of as a generalization of the concept of position).
Technically, the symplectic form is defined as a closed, nondegenerate, -form. By “-form“, we mean that it is a differential form, obeying the properties we gave in Differential Forms, such as antisymmetry. The notion of a differential being “closed“, also already discussed in the same blog post, means that its exterior derivative is zero. “Nondegenerate” of course was already defined in the preceding paragraphs. The symplectic form is also a bilinear form, although this is a property of all -forms, considered as functions of two tangent vectors at some point on the manifold. More generally, all differential forms are examples of multilinear forms. A manifold with a symplectic form is called a symplectic manifold.
There is still so much more to differential geometry, but for now, we have at least accomplished the task of defining some of its most basic concepts in a more technical manner. The language we have discussed here is important to deeper discussions of differential geometry.
Differential Geometry of Curves and Surfaces by Manfredo P. do Carmo
Differential Geometry: Bundles, Connections, Metrics and Curvature by Clifford Henry Taubes
Foundations of Differential Geometry by Shoshichi Kobayashi and Katsumi Nomizu
Geometry, Topology, and Physics by Mikio Nakahara