Understanding Coordinate Frames on Manifolds

[Tedjn]

A little embarrassing, but I have had very little exposure to anything involving manifolds and am trying to work through these notes over spring break. I will have many questions on even the simplest concepts. In this thread I hope to outline these as I encounter them, and if anyone can help I would be very grateful.

Even in the beginning of the preliminary, I am encountering some difficulties. In the definition of coordinate frame associated with a chart x, the notes say it is a tuple of vector fields. What are these vector fields given by the partial derivative notation as applied to a general differentiable manifold? How many are there, n of them? Is each defined over U, the domain of x? And of course, is there some better intuitive way to understand this frame concept that I haven't seen?

These are some questions to start off just in the first paragraph, so you see what level I am at. Thanks in advance for any insights :)

 

[Kreizhn]

I looked over these notes, and I think the assumption is really that you already have a decent background in differential geometry. You might want to try learning some general smooth manifold theory before moving on to Kahler manifolds.

I would suggest Introduction to Smooth Manifolds, by John Lee. It's a fantastic book that I've read cover to cover and continually go back to as a reference.

 

[Kreizhn]

In light of my previous post, I will attempt to answer your question.

A big goal of differential geometry is to be able to do everything in a coordinate independent way. We don't want to have to depend on whether we're working with "Cartesian" or "Polar" coordinates or any such nonsense. Similarly, we want to avoid thinking about any particular embedding of a manifold. So for example, when you think of the 2-sphere, you probably think of it as a subspace of $\mathbb R^3$. The point is that you need to let go of that thinking as see the 2-sphere as something that exists without a choice of coordinates or embedding.

As for your particular problem, if we ever want to actually do computations with manifolds, it is an unfortunate reality that we must define a coordinate system. So in particular, if M is your manifold, choose a point $p \in M$ and a chart $(U,\phi)$ for some neighbourhood U of p and $\phi : U \to V \subseteq \mathbb R$. Define a local coordinate system on V, which we can pull back to M.

Now at p we can define the tangent space $T_p M$. There are many equivalent ways of defining this space, though my favourite is as the set of derivations at p. That is, $T_p M$ consists of all functions $X_p: C^\infty(M) \to \mathbb R$ that satisfy the Leibniz rule
$$X_p(fg) = X_p(f)g(p) + X_p(g) f(p)$$
Notice that this definition is invariant of any coordinate system. However, if we have a set of coordinates $\{ x_i \}$ in a neighbhourhood of p, we can lift them to coordinates on the tangent space
$$\left\{ \left. \frac{\partial}{\partial x_i } \right|_p \right\}$$
Note that you have to be careful about how these partials work, in particular
$$\left. \frac{\partial}{\partial x_i } \right|_p = d(\phi^{-1}) \left. \frac{\partial}{\partial x_i } \right|_{\phi(p)}$$
where the right hand side represents the usual partials in $\mathbb R^n$ and d represents the pushforward/differential operator, which maps tangent vectors between spaces.

Now we define the tangent bundle as
$$TM = \bigcup_{p \in M} T_p M$$
where the union is taken in a disjoint fashion. Then there is a natural projection $\pi: TM \to M$ where $\pi(V,p) = p$. Then vector fields are sections of $\pi$. That is, they are functions $X: M \to TM$ such that $\pi \circ X = \text{Id}_M$ the identity map on M.

Then a local frame at p is a set of vector fields $\{ X_i \}$ such that their evaluation at q gives a basis for the tangent space $T_q M$ for all q in a neighbourhood of p. Since we've specified a coordinate basis, we can then write each basis as
$$X_i(p) = X_i^j \left. \frac{\partial}{\partial x^j} \right|_p$$

Hope that helps.