A urban digital twin is the virtual representation of real assets, processes, systems and subsystems of a city. It uses and integrates heterogeneous data to learn and evolve with the physical city, providing support to monitor the current status and predict/anticipate possible future scenarios. In this paper, we focus on the issues and potential related to the geometric layer of the city digital twin. On the one hand, detailed 3D data to reconstruct the urban morphology very accurately might not be available, and planning a new survey is costly in terms of money and time. On the other hand, the more the geometry adheres to the real counterpart, the more accurate measures and simulations related to the urban space will be. We describe our approach to develop the geometric layer of the digital twin of the city of Matera, in Italy, using only pre-existing public data. Specifically, our method exploits available digital elevation models from a previous regional aerial survey and integrates them with data coming from OpenStreetMap to generate an as-precise-as-possible 3D model, annotated with heterogeneous semantic information. We demonstrate the potential of the geometric layer by developing two geometric characterisation services, namely route slope extraction and light/shadow maps according to a specific date and time. In the next steps, the computed attributes will help to answer specific objectives which could be of interest for the Municipality, such as personalised optimal routes taking into account user preferences including slope and perceived environmental comfort.