It is $\mathbb{D} = \{ u \in \mathbb{C} \mid |u|<1 \}$ with a Riemannian metric. It is a model for the hyperbolic geometry, similar to the Poincare half plane, but now the horizontal line of the infinity together with the infinity point common to all the vertical lines is rounded to the boundary of the disk. See [Needham 2021]. In the note Poincare half plane#Map to the Poincare disk there is a map that carry $\mathbb{H}$ to $\mathbb{D}$ with their respective metrics.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es