For the Poincaré half-space model in dimension 2, the metric evaluates on the coordinate tangent vectors \(\frac{\partial}{\partial x}, \frac{\partial}{\partial y} \in T_pM\) as \[g_p\!\left(\frac{\partial}{\partial x^i}\bigg|_p,\;\frac{\partial}{\partial x^j}\bigg|_p\right) = \frac{1}{y^2}\,\delta_{ij},\] i.e. the coordinate tangent vectors are orthogonal and each has length \(\frac{1}{y}\) — shrinking to zero as \(p\) approaches the boundary \(y\to 0\), which is what makes the space “infinitely large” near the boundary.
Everybody needs good neighbours. Photograph: Harold M Lambert/Getty ImagesThe disappearance of these kinds of interactions from day-to-day life – in pubs, restaurants, shops, queues, on public transport – is striking. I have been talking to people tangentially about this for the past 10 years, ever since I started researching my book, How to Own the Room, which came out in 2018 and went on to become a podcast. This project was supposed to be about public speaking and confidence. But I realised from people’s reactions to the topic – especially younger people – that their deepest anxiety lies elsewhere, in something much more banal and inexpressible. Forget “public speaking”. What a lot of people don’t like at all any more is “speaking to anyone in public”.
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Was it to show support for Iran’s opposition, as Trump’s earlier “HELP IS ON ITS WAY” posts on Truth Social suggested? A bombing campaign that was bound to kill innocent Iranians, including 175 people at a girls elementary school near a military base, seemed an odd form of support.
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