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I was just reading about the hyperbolic space (upper-half plane model) and i'm getting kind of confused about the notation for the Riemannian metric. The half-plane is defined as $$ H = \{(x,y) \in \mathbb{R}^2 | y > 0\}, $$ and the metric is defined as $$ ds^2 = \frac{dx^2 + dy^2}{y^2}. $$ I've just taken a first course in Riemannian geometry and I have absolutely no idea where this notation comes from and what it means. We usually defined metrics explicitely as some bilinear map $g$. Thanks for your time.

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At each point $(x,y)\in H$, the tangent space $T_pH$ at $p$ is nothing but spanned by $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$, i.e. every tangent vector at $p$ is in the form of $a\frac{\partial}{\partial x}+b\frac{\partial}{\partial y}$ for some $a,b\in\mathbb{R}$.

With this understanding, the Riemannian metric $g=ds^2 = \frac{dx^2 + dy^2}{y^2}$ means the following $$g(\frac{\partial}{\partial x},\frac{\partial}{\partial x})=\frac{dx^2(\frac{\partial}{\partial x},\frac{\partial}{\partial x})}{y^2}=\frac{1}{y^2}$$ and $$g(\frac{\partial}{\partial y},\frac{\partial}{\partial y})=\frac{dy^2(\frac{\partial}{\partial y},\frac{\partial}{\partial y})}{y^2}=\frac{1}{y^2}$$ and $$g(\frac{\partial}{\partial x},\frac{\partial}{\partial y})=0$$ since there is no $dxdy$ term. And these are extended linearly, i.e. $$g(a\frac{\partial}{\partial x}+b\frac{\partial}{\partial y},c\frac{\partial}{\partial x}+d\frac{\partial}{\partial y})=ac\,g(\frac{\partial}{\partial x},\frac{\partial}{\partial x})+(ad+bc)g(\frac{\partial}{\partial x},\frac{\partial}{\partial y})+bd\,g(\frac{\partial}{\partial y},\frac{\partial}{\partial y})=...$$

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