Preliminaries to Studying Non-Euclidean Geometry

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Introduction

Continuum mechanics applies the tools of differential geometry to study stress and strain fields within a material by treating that material as an idealized continuum. In most cases, we assume that we assume Euclidean geometry for the continuum, but there are several situations when the assumptions of Euclidean geometry no longer apply.

One example is using the tools of continuum mechanics at astronomical length-scale where the effects of gravity, i.e. the curvature of space, become non-negligible, such as when studying neutron stars. [1] [2]

Another example is when applying continuum mechanics at the mesoscale to study material defects such as dislocations, voids and disclinations. [3] [4] [5] Normally, the continuum idealization would have meant that we cannot model explicitly discontinuities in the material such as due to dislocations and disclinations, but as the aforementioned references show, this problem can be overcome by applying non-Euclidean geometry to the discontinuous material.

  • Potential (Reasons to pursue this further): Non-Euclidean computational models: could be more accurate as they will be able to naturally handle discontinuities and incompatibilities.
  • Challenges: Not part of the typical continuum mechanics curriculum. Few engineers are trained in this. Non-Euclidean geometry is often veiled in highly abstracted mathematical language (tangent bundles, vector fields, Lie groups and lie algebra, differential forms, ... ). While some of these abstractedness is necessary to ensure that we are not overly limiting or overly extending the applicability of the math, it is often more helpful to first grasp the basic concepts intuitively and then the generalization will become more clear.
  • Here we will focus on an attempt to intuitively present the most basic abstractions in a way that would hopefully lend itself to better intuitive and not just syntactic understanding.

Preliminary concepts: Metric (like a measuring stick), Co-variant and Contra-variant vectors (different representations of a vector once we have have a metric defined), Geometric connections: (how can we compare vectors at different points in space),

Metric and the Metric Tensor

TODO

Covariant and Contravariant Vector Representations

TODO

Geometric Connections and Co-variant Derivative

TODO

Materials for Further Study

TODO


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