A sheet of paper is intrinsically flat even when it is bent, but a spherical surface is intrinsically curved. The general theory of intrinsically curved spaces was originally developed by Gauss and Riemann, and during the last century, theoretical physics found a striking use of this theory in Einstein's General Relativitity, where one considers not just spatial curvature, but the curvature of space and time together. While the formalism of curved spaces and general relativity have been simplified over time and can now be taught to advanced undergraduates, for most people, daily experience and intuitive understanding of intrinsically curved spaces is limited to the simplest case of two-dimensional surfaces.
In this talk, I will describe a somewhat surprising, "everyday" appearance of 3-dimensional curved spaces in the field of continuum mechanics. The context where geometry appears is somewhat different from that of general relativity, where it is the 'background space' that is curved. I will begin by giving elementary--hopefully undergraduate-level--introductions to thermoelasticity and Riemannian geometry, and continue with the geometric equations we propose as an alternative foundation to thermoelasticity. We will interpret certain puzzling aspects of the classical theory in the context of our approach, and mention other intriguing possibilities in the theory of biological materials and growth, which are still under investigation.