Riemannian metric, compute I have a question that may look for you as silly. A few years ago I took a course of Riemannian geometry. Well, the first problem I found is to understand the generalization of tangent plane (in surfaces) into tangent space. I accept the idea (caused by my confusion) and then we start with riemannian metric, in the definition they use
$g_{i,j}=\langle\frac{\partial}{\partial x_i},\frac{\partial}{\partial x_j}\rangle$, but then when they (the authors) start with examples, they give for the hyperbolic plane
$g_{11}=g_{22}=\frac{1}{y^2}$. First, in the definition, what means $\langle,\rangle$? Second, how do we get for the hyperbolic plane that $g_{11}=g_{22}=\frac{1}{y^2}$ by computing that "inner product"? Third, how do we get from this an expresion for the inner product betwen two tangent vectors in the hyperbolic plane? Notice, what I am trying to understand is this in the general case, the hyperbolic plane is just an example. Finally, how do I start to compute metric and inner product in the projective plane?
Thanks for your help, and I apologize for my english.
 A: You first have to think about linear algebra. Consider a finite-dimensional real vector space $V$ with a basis $e_1,...,e_n$. I assume you already know what a positive-definite inner product on $V$ is; the inner product of vectors $u$ and $v$ is usually denoted by 
$$
\langle u, v\rangle. 
$$ 
I assume also that you remember from your linear algebra that each inner product corresponds to a certain positive-definite symmetric $n\times n$ matrix, called the Gramm matrix $G$, whose entries are 
$$
G_{ij}= \langle e_i, e_j\rangle.  
$$
Conversely, each positive-definite symmetric $n\times n$ matrix is the Gramm matrix of some inner product on $V$ (I keep the basis fixed). I will denote the space of such matrices by $Sym_+(V)$ (it is an open subset of the space of all symmetric matrices). Thus, one can use matrices in order to define inner products. 
Once you remember all this, you can think of a Riemannian metric on an open subset $U$ of ${\mathbb R}^n$ as a smooth function 
$$
g: U\to Sym_+({\mathbb R}^n). 
$$
Here, as the basis in each tangent space $T_x U$ we use the standard basis, which differential geometers like to denote 
$$
e_i= \frac{\partial}{\partial x_i}
$$
since they prefer to identity tangent vectors and directional derivatives. (Why - is another story.) But as a practical matter, to define a Riemannian metric you just have to specify 
$$
\frac{n(n+1)}{2}
$$
smooth functions $g_{ij}(x)$ on $U$ (entries of your matrix-valued function). 
(Why don't I need $n^2$ functions?) For instance, the standard Riemannian metric on ${\mathbb R}^n$ is given by the constant functions  $g_{ij}=\delta_{ij}$, i.e., by the identity n-by-n matrix $I_n$. 
Now, we can make sense of the hyperbolic metric. I will use the coordinates $x_1, x_2$ on the upper half-plane so that $x_2>0$. Then the expression you have simply means that the metric is given by the function
$$
(x_1,x_2)\mapsto \left[\begin{array}{cc}
x_2^{-2}&0\\
0&x_2^{-2}\end{array}\right]=  x_2^{-2} I_2.
$$
For instance, if you want to compute the inner product of $e_1$ and $e_1+e_2$ in the tangent space at the point $(x_1,x_2)=(0,2)$, you simply take their standard inner product (I assume, you know how to compute it) and divide it by $x_2^2=4$. 
Now, defining and computing inner product on n-dimensional manifolds which are not open subsets of ${\mathbb R}^n$ is more difficult. One way to do so is to define Riemannian metrics on each $U_\alpha\subset {\mathbb R}^n$ for a chart
$$
\phi_\alpha: U_\alpha\to M
$$
and make sure that the transition maps $\phi_\beta^{-1} \circ \phi_\alpha$ preserve the metric. Another option is to embed your manifold $M$ in some ${\mathbb R}^N$ via a (smooth) embedding $f: M\to {\mathbb R}^N$ and then take the pull-back metric
$$
g=f^*(g_e)
$$
where $g_e$ is the standard Riemannian metric on ${\mathbb R}^N$. How to do so, is another story, but I would not want to write a differential geometry textbook here, at MSE, which is not intended for such activity (and I have no desire to write one anyhow!). 
A: There are two concepts here that are kind of at odds with each other.
When you have a manifold embedded in a larger ambient manifold, the metric of the larger manifold induces a metric on the embedded one.  In the case of a surface embedded in 3d space, a parameterization of the surface generates tangent vectors at each point on the surface, and the inner products of those tangent vectors--using the ambient space's metric--determine the metric on the surface.  A good exercise here would be finding the metric of an embedded 2-sphere using the spherical coordinates.
If there is no embedding, however, the metric has to be given to you (or perhaps, calculated from other information, but more often than not it should just be given to you).
The expression $g_{ij} = \langle \partial_i, \partial_j \rangle$ is just a way of saying that the metric is what you use to compute inner (or dot) products.  If the manifold is embedded, you can use the higher-dimensional manifold's metric to compute the inner product, and from there you can compute the embedded manifold's metric components.  Otherwise, though, you have to be given the components $g_{ij}$ just to begin to compute inner products of tangent vectors.
A: Well, this might give you some insight. 
Let $(M, g)$ be a Riemannian manifold. Let us write $g$ locally. Let $p\in M$ and take $(U, x_1, \ldots, x_n)$ a chart around $p$. The Riemannian metric $g$ is a map which associates to each $p\in M$ an inner product $$g_p:T_pM\times T_pM\longrightarrow \mathbb R.$$ This kind of map is best understood if you know something about vector bundles for it is symply a section of a given bundle. 
Now, take $X_p, Y_p\in T_pM$. Then we may write $$X_p=\sum_{i=1}^n a_i(p)\frac{\partial}{\partial x_i}\biggr|_p\quad \textrm{and}\quad Y_p=\sum_{i=1}^n b_i(p)\frac{\partial}{\partial x_i}\biggr|_p.$$ By the bilinearity of $g_p$ we get 
$$g_p(X_p, Y_p)=\sum_{i, j=1}^n a_i(p)b_j(p)g_p\left(\frac{\partial}{\partial x_i}\biggr|_p, \frac{\partial}{\partial x_j}\biggr|_p\right)=\sum_{i, j=1}^n g_{ij}(p)((dx_i)_p\otimes (dx_j)_p)(X_p, Y_p)$$ where $$g_{ij}(p)=g_p\left(\frac{\partial}{\partial x_i}\biggr|_p, \frac{\partial}{\partial x_j}\biggr|_p\right)$$ and $(dx_i)_p$ is the map that takes $X_p\in T_pM$ and extracts the $i$-th coordinate of $X_p$. Since the above holds for all $(X_p, Y_p)\in T_pM\times T_pM$ we find $$g_p=\sum_{i, j=1}^n g_{ij}(p)(dx_i)_p\otimes (dx_j)_p.$$ Therefore, a riemannian metric has the form $$g=\sum_{i, j=1}^n g_{ij}dx_i\otimes dx_j.$$ So, $g_{ij}$ is nothing but a matrix $n\times n$ whose entries are functions on $M$. For example, in a point $p\in M$, $g_{ij}(p)$ is the matrix whose entries are given by $$g_p\left(\frac{\partial}{\partial x_j}\biggr|_p, \frac{\partial}{\partial x_j}\biggr|_p\right).$$ Note $g_{ij}(p)$ is symply an scalar for each $i, j=1, \ldots, n$ for $g_p:T_pM\times T_pM\longrightarrow \mathbb R$. 
The fact $g_p$ is a real inner product tells us $g_{ij}(p)$ is in fact a symmetric matrix and that it is positive definite for every $p\in M$. 
Regarding your first question, if you read carefully the above construction, you will see clearly what the notation $\langle, \rangle $ means. 
