I've started reading Kazaryan's "Calculus on Manifolds", to say at start, I have some intuition on manifolds, charts and atlases, but haven't really taken any smooth manifold classes, thus started reading some booklets on given topic. But have stuck almost in the beginning. And here is why. In the book, the author gives a definition of a k-dimensional submanifold in $\mathbb{R}^n$:
A $k$-dimensional submanifold in $\mathbb{R}^n$ is a set $M \subset \mathbb{R}^n$ s.t. each point $x \in M$ has a nbhd $U_x \subset \mathbb{R}^n$ in which $M$ is determined by a system of equations $f_1(x) = … = f_{n-k}(x) = 0$, where all the functions $f_i$ are smooth and the matrix of partial derivatives $(\frac{\partial f_i}{\partial x_j})$ has maximal possible rank $n – k$.
For example, consider $S^1$. Obviously, this is a manifold. Do I get the definition above correct if I say that $S^1$ is determined via equation $x^2 + y^2 – 1 = 0$?
Further, the author introduces the definition of local coordinates:
A set of functions $y_1, …, y_k: \mathbb{R}^n \to \mathbb{R}$ is a set of local coordinates in a nbhd of $x \in M \subset \mathbb{R}^n$, where M is a k-dimensional submanifold given by a system of equations $f_1 = … = f_{n-k} = 0$, if the functions $y_1, …, y_k, f_1 = … = f_{n-k}$ together determine a system of local coordinates in this nbhd of $x \in \mathbb{R}^n$, i.e. the matrix of the 1st partial derivatives of the set $y_1, …, y_k, f_1, …, f_{n-k}$ is nondegenerate at $x$. In other words, the local structure of a $k$-submanifold coincides with that of $k$-space.
Reading this, there is a picture of a, say, plane in $\mathbb{R}^3$, i.e., determined by the equation $z = 0$ and local coordinates $(x,y,z) \mapsto x,\; (x,y,z) \mapsto y$. This seems to be correct and the given definitions work fine. But if consider $S^1$ again, I don't see such picture in this case. If my assumption, that $S^1$ is determined by the equation $x^2+y^2-1=0$ is correct, what are then local coordinates? I would say that, e.g. an angle between $x$-axis and radius are local coordinates in the sense that I can code any point on $S^1$ by that, but I don't really see how I can connect that with the definition of local coordinates given above.
P.S. Can someone recommend a good, but not too thick book(e.g. Lee — Introduction to smooth manifolds) on differential geometry, that won't be too narrow, but still cover everything, that a proper math grad student should know? Thanks!
Best Answer
To address your mathematical questions:
Yes, the unit circle in the Cartesian $(x, y)$-plane may be defined by $f_{1}(x, y) = x^{2} + y^{2} - 1 = 0$. The matrix of partial derivatives, $Df_{1}(x, y) = [2x\quad 2y]$ has (maximal) rank $1$ at each point of the circle.
Local coordinates for the circle in the sense of your second quoted passage are the independent variables if we solve the implicit equation for one variable in terms of the other. Thus, $x$ serves as a local coordinate on the set of points where $y > 0$, intuitively because we have $y = \sqrt{1 - x^{2}}$ for $-1 < x < 1$, and the portion of the unit circle where $y > 0$ is precisely the set of pairs $(x, \sqrt{1 - x^{2}})$ for which $-1 < x < 1$. Similarly, $x$ serves as a local coordinate on the set of points where $y < 0$ because $y = -\sqrt{1 - x^{2}}$ for $-1 < x < 1$. Analogously, $y$ serves as a local coordinate on the set of points where $x > 0$, or where $x < 0$.
On the other hand, $y_{1} := x$ does not serve as a local coordinate in a neighborhood of $(1, 0)$; consistent with this, the derivative matrix of the mapping $(y_{1}, f_{1})$ is degenerate (rank $1$) at $(1, 0)$. Similar remarks hold at the other three "cardinal points".
The definition of a manifold as a topological space locally modeled by open subsets of a Cartesian space is related, but manifolds as in the book you're reading are equipped in addition with an embedding into Cartesian space.
Local coordinates, moreover, are assumed in your book to be restrictions of Cartesian coordinates. A choice of polar angle on the circle is not a local coordinate in this sense.
You can think of an embedding as distinguishing a finite number of coordinate systems near each point of a manifold, those obtained by restricting some choice of $k$ Cartesian coordinates. There may be as many as $\binom{n}{k}$ such choices, but at points where the tangent space contains a line perpendicular to a coordinate axis (i.e., parallel to some coordinate hyperplane) there are fewer.