Hi I have been trying to understand how to see the gradient of a function $f:M\to \mathbb{R}$, where $M$ is a Riemannian manifold. Obviously to do this I need to have a firm understanding of then tangent space, etc.
I have done some reading, but haven't come across the 'quintessential example' of a Riemannian manifold. The examples here https://en.wikipedia.org/wiki/Riemannian_manifold#Examples I don't find very illustrative. I was expecting to find a sphere or some other shape as a subset of $\mathbb{R}^n$.
What examples of Riemannian manifolds and their tangent spaces do you find most enlightening?
I'll write a short list, just for concreteness. Dealing with tangent spaces does not require a metric at all.
if $A$ is an affine space with translation vector space $V$, then $T_pA = V$ for all $p \in A$. An inner product in $V$ defines a Riemannian metric on $A$. This is a mild generalization of what people do in vector spaces, as we "forget the origin".
$\Bbb S^n =\{ x \in \Bbb R^{n+1} \mid \|x\|=1$, and we have $T_p(\Bbb S^n)= x^\perp$ for all $p \in \Bbb S^n$. This inherits a Riemannian metric from $\Bbb R^{n+1}$. I don't think I need to explain why spheres are relevant.
if $V$ is a vector space, then the collection ${\rm Gr}_k(V)$ of all $k$-dimensional vector subspaces of $V$ is a manifold and $T_W{\rm Gr}_k(V) \cong {\rm Hom}(W,V/W)$. For $k=1$, this is the projective space over $V$. It not as straightforward, but this also inherits a Riemannian metric from any inner product in $V$ (keyword: Fubini-Study metric). These are a nice source of examples, especially in the complex case (where we obtain compact homogeneous Kähler-Einstein manifolds).
if $V$ is a vector space with an indefinite scalar product, and $\mathcal{C} =\{x \in V \setminus \{0\}\mid \langle x,x\rangle=0$ is the lightcone of $V$, then $T_x\mathcal{C} = x^\perp$. If we define $\sim$ on $\mathcal{C}$ by saying that $x\sim y$ if $y=\lambda x$ with $\lambda \neq 0$, then the quotient $\mathbb{E}=\mathcal{C}/_{\sim}$ is also a manifold (it's the collection of all lightrays in $V$) and $T_L\mathbb{E} = x^\perp/\Bbb R x$, where $x$ is any vector with $L = \Bbb R x$. These do not inherit Riemannian or pseudo-Riemannian metrics from $V$. The scalar product restricted to $\mathcal{C}$ is degenerate, and what survives in the quotient $\mathbb{E}$ is just a conformal structure, which turns out to be conformally flat (to wit, $\mathbb{E}$ is diffeomorphic to a product of an anti-Euclidean sphere -- with negative-definite metric --- and an Euclidean sphere, whose dimensions are the positive and negative indices of the indefinite scalar product in $V$, both subtracted by $1$). This $\mathbb{E}$ is called "the Einstein manifold" (and is unrelated to the notion of Einstein manifolds, where the Ricci tensor is a multiple of the metric).
If $G$ is a Lie group, then $T_gG = {\rm d}(L_g)_e[\mathfrak{g}]$, where $\mathfrak{g}$ is the Lie algebra if $G$. In particular, you get things such as $$T_S{\rm SO}(n) = \{SA \in \Bbb R^{n\times n} \mid A \mbox{ is skew-symmetric}\},$$etc. Any inner product in $\mathfrak{g}$ gives rise to a left-invariant Riemannian metric on $G$, via left translations. In particular, metrics in ${\rm SO}(3)$ of the form $$\langle\!\langle V,W\rangle\!\rangle = \int_{\Bbb R^3} \langle V\xi,W\xi\rangle\,{\rm d}\mathfrak{m}(\xi),$$where $\mathfrak{m}$ is a finite measure not supported in any line through the origin, are used in the study of rigid bodies, in classical mechanics.