Understanding Coordinate Frames on Manifolds

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In summary, the conversation discusses the concept of coordinate frames associated with a chart on a manifold and the difficulties in understanding them. The goal of differential geometry is to be able to work in a coordinate-independent way, but for computations, a coordinate system must be defined. The tangent space is defined as a set of derivations at a point and the tangent bundle is the union of tangent spaces at all points. Vector fields are defined as sections of the tangent bundle and a local frame is a set of vector fields that form a basis for the tangent space.
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Tedjn
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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 :)
 
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  • #2
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.
 
  • #3
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 [itex] \mathbb R^3 [/itex]. 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 [itex] p \in M [/itex] and a chart [itex] (U,\phi) [/itex] for some neighbourhood U of p and [itex] \phi : U \to V \subseteq \mathbb R[/itex]. Define a local coordinate system on V, which we can pull back to M.

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

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

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

Hope that helps.
 
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1. What are manifolds?

Manifolds are mathematical objects that can be described as surfaces or curved spaces that locally resemble open subsets of Euclidean space. They are commonly used in many fields of mathematics and physics to model complex systems.

2. Why is learning about manifolds important?

Understanding manifolds is important because they provide a powerful framework for studying and solving problems in various fields such as geometry, topology, and physics. They also have practical applications in computer science, machine learning, and data analysis.

3. What are the key concepts in learning about manifolds?

Some key concepts in learning about manifolds include dimensionality, coordinate charts and atlases, smoothness, tangent spaces, and differential forms. These concepts help us define and understand the properties and behavior of manifolds.

4. How are manifolds different from other mathematical objects like vectors or matrices?

Manifolds are different from other mathematical objects in that they are not defined by a set of coordinates or numbers. Instead, they are defined by a set of rules or equations that describe the behavior and properties of the manifold. Additionally, manifolds can have different dimensions and can be curved, unlike vectors or matrices which are typically linear and have fixed dimensions.

5. What are some real-world applications of manifolds?

Manifolds have many real-world applications, such as in robotics for motion planning, in computer graphics for creating smooth surfaces, and in physics for understanding the behavior of space-time. They are also used in data analysis for dimensionality reduction and in machine learning for data classification and clustering.

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