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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.
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"Tuning frictional behavior on the fly has been a long-standing engineering dream," said co-author Katia Bertoldi of Harvard University. "This new insight into how surface geometry governs slip pulses paves the way for tunable frictional metamaterials that can transition from low-friction to high-grip states on demand.” In addition, the dynamics revealed by these results are similar to those of tectonic faults and thus give scientists a new model for the mechanics of earthquakes, according to their new paper published in the journal Nature.