I'm studying differentiable manifolds with the book Warner. "Foundations of Differentiable Manifolds and Lie Groups." It defines differentiable structure as follows but I think maybe the connectedness condition is included mistakenly.

1.3 Definitions $\;$ A locally Euclidean space $M$ of dimension $d$ is a Hausdorff topological space $M$ for which each point has a neighborhood homeomorphic to an open subset of Euclidean space $\mathbb R^d$. If $\varphi$ is a homeomorphism of a connected open set $U\subset M$ onto an open subset of $\mathbb R^d$, $\varphi$ is called a coordinate map, the functions $x_i = r_i \circ \varphi$ are called the coordinate functions, and the pair $(U, \varphi)$ (somethimes denoted by $(U, x_1, \ldots, x_d)$) is called a coordinate system. (The rest omitted because it is irrelavent to my question.)

1.4 Definitions $\;$ A differentiable structure $\mathcal F$ of class $C^k$ $(1\leq k\leq \infty)$ on a locally Euclidean space $M$ is a collection of coordinate systems $\{(U_\alpha, \varphi_\alpha): \alpha\in A\}$ satisfying the following three properties:
(a) $\bigcup_{\alpha\in A}U_\alpha = M$.
(b) $\varphi_\alpha\circ\varphi_\beta^{-1}$ is $C^k$ for all $\alpha,\beta\in A$.
(c) The collection $\mathcal F$ is maximal with respect to (b); that is, if $(U,\varphi)$ is a coordinate system such that $\varphi\circ\varphi_\alpha^{-1}$ and $\varphi_\alpha\circ\varphi^{-1}$ are $C^k$ for all $\alpha\in A$, then $(U,\varphi)\in\mathcal F$.

And here is the definition of differentiable manifold in the book.

A $d$-dimensional differentiable manifold of class $C^k$ is a pair $(M, \mathcal F)$ consisting of a $d$-dimensional, second countable, locally Euclidean space $M$ together with a differentiable structure $\mathcal F$ of class $C^k$.

My question is: Is it standard to put connectedness condition in definition 1.3? Becuase the book seems to ignore the connectedness condition in the texts that follows the above definitions (I'm not sure though, because I only read very little). For example, see the following.

An open subset $U$ of a differentiable manifold $(M,\mathcal F_M)$ is itself a differentiable manifold with the differentiable structure $\mathcal F_U = \{(U_\alpha\cap U,\varphi_\alpha|U_\alpha\cap U):(U_\alpha,\varphi_\alpha)\in\mathcal F_M)$.

Isn't this incorrect if there is a connectedness condition in definition 1.3?

  • $\begingroup$ The condition of connectedness is redundant for the definition of manifold, but perhaps Warner finds it convenient for his "coordinate maps" to have connected image. $\endgroup$ – Angina Seng Jan 7 '19 at 5:34
  • $\begingroup$ More importantly, is the condition of connectedness restrictive? i.e. are there manifolds with unconnected charts? $\endgroup$ – stressed out Jan 7 '19 at 6:15
  • $\begingroup$ @Lord Why is it redundant? If there were no connectedness condition, can't $(\mathbb R, \mathcal F)$ be a differentiable manifold, where $\mathcal F$ is a differentiable structure containing $\varphi_1: \mathbb R-\{0\}\to\mathbb R-\{0\}, x\mapsto x$ and $\varphi_2: \mathbb R\to\mathbb R, x\mapsto x$? Having a connectedness condition will not allow $\varphi_1$ to be in $\mathcal F$ $\endgroup$ – zxcv Jan 7 '19 at 7:37
  • $\begingroup$ @stressed Isn't it restrictive, as in my above comment? $\endgroup$ – zxcv Jan 7 '19 at 7:39
  • $\begingroup$ @zxcv $\Bbb R$ is a differentiable manifold. $\endgroup$ – Angina Seng Jan 7 '19 at 7:46

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