One can find many pictures to illustrate examples of the tangent bundle, eg from wiki
The simple explanation would be that, to get the tangent bundle, you collect union up the tangent spaces at each point of the circle. Simple, intuitive and straight to the point.
But, how does the cotangent space looks like? If not a picture, could some geometric intuition for it be provided?
I couldn't find any good sources on this after searching :/
On
$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\cW}{\mathcal{W}}$ I think it is helpful to consider the cotangent space/bundle in the context of a pairing. If a vector space $V$ is embedded in $\R^n$, then any subspace $W$ of the same dimension with $V$ and does not contain a vector orthogonal to $V$ could be identified with the dual space of $V$, with the pairing $(w, v)\mapsto w^Tw = w.v$. So when $V$ is one dimension, any line through $0$ slanted to $V$ could be considered a dual space of $V$ in this pairing.
For a manifold $M\subset \R^n$ (we will work with embedded manifolds to help the intuition), the requirement is to have a bundle $\cW$ such that at each $x\in M$, $\cW_x$ cannot contain any vector orthogonal to the whole of $T_xM$, and each $\cW_x$ has the same dimension as $T_xM$. In particular, $TM$ could be identified with $T^*M$, but as in the case of linear algebra, there is more than one possible choice of $\cW$.
This requirement is equivalent to the existence of an affine projection function $\Pi$, each $\Pi(x)$ is a matrix in $\R^{n\times n}$, $\Pi(x)^2=\Pi(x)$, and the range of $\Pi(x)$ is exactly $T_xM$. Then the range of the transpose matrix $\Pi(x)^T$ could be identified with $T_x^*M$, and we can construct such a projection $\Pi$ for any subbundle $\cW$ of $M\times\R^n$ with a nondegenerate pairing. This is done in [2]. We use this description to provide an explicit description of Hamilton vector fields for embedded manifolds, extending [1].
Two examples: If $M$ is a mass-metric operator, ie $M(x)$ is a positive definite matrix in $\R^{n\times n}$ for each $x\in M$ and $M$ is smooth as a function of $x$, then one construction is to identify $T^*_xM$ with $M(x)T_xM$, inspired by the momentum formula $p = m.v$. Note that $M(x)T_xM$ may be different from $T_xM$, as we can take $M(x)$ to be a constant diagonal matrix, for example, in the case of the sphere. The projection $\Pi$, in this case, is the projection compatible with the metric induced by $M$ $$M(x)\Pi(x) = \Pi(x)^TM(x). $$ and there are simple formulas for $\Pi$.
For the unit sphere $S^{n-1}\subset \R^n$ with the constraint $x^Tx = 1$, another construction is to use the projection $$\Pi_{a}(x):\omega\mapsto \omega - (x + a)\frac{x^{T}\omega}{1+x^{T} a} $$ where $a\in \R^n$ has norm $|a| < 1$, identifying $T_x^*M$ with the subspace orthogonal to $x+a$, with $a=0$ identifying $T^*M$ with $TM$.
[1] T. Lee, M. Leok, and N. Harris McClamroch, Global formulations of Lagrangian and Hamiltonian dynamics on embedded manifolds, Proceedings of the IMA Conference on Mathematics of Robotics, 2015
[2] Nguyen, D. Geometry in global coordinates in mechanics and optimal transport https://doi.org/10.48550/arXiv.2307.10017
Each red line in the picture represents what we call the fiber of a bundle. There is exactly one for each point in the base manifold (depicted as the blue circle), commonly denoted $M$.
Each type of bundle has a specific type of fiber. The tangent bundle fiber at $p\in M$ is called the tangent space at $p$ and denoted $T_pM$. This is defined rigorously in all the standard textbooks on manifolds or differential geometry. For example, it can be viewed as the space of all possible velocity vectors at $p$ of curves passing through $p$.
From here, it’s linear algebra. $T_pM$ turns out to be a vector space. There are other vector spaces naturally associated to a vector space $V$. The most important is the dual vector space $V^*$. The cotangent bundle is the bundle whose fiber at $p$ is the dual vector space of $T_pM$, denoted $T^*M$.