Abstract
In the 1820s Gauss showed how one can do fundamental geometry on a two-dimensional surface independent of how the surface might be contained in an ambient space. This involved formulating a metric on the surface and a way of measuring angles, curvature, and other geometric objects in an intrinsic way on the surface. It formed the basis for what became the discipline of di erential geometry in the twentieth century.
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- 1.
“Investigations of curved surfaces”.
- 2.
Gauss used here simply \(z=z(x, y)\), and we have modified his notation in this one instance for clarity.
- 3.
We note that the three representations of curvature in Sect. 4.1 depend on first and second derivatives of the defining functions for the surface, whereas the Gaussian curvature in (4.10) depends on three derivatives of the defining functions. This comes about since the change of variables in (4.9) involves first derivatives.
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“These theorems lead us to consider the theory of curved surfaces from a new point of view, whereby the investigations open to a quite new undeveloped field. If one doesn’t consider the surfaces as boundaries of domains, but as domains with one vanishing dimension, and at the same time as bendable but not as stretchable, then one understands that one needs to differentiate between two different types of relations, namely, those which assume the surface has a particular form in space, and those that are independent of the different forms a surface might take. It is this latter type that we are talking about here. From what was remarked earlier, the curvature belongs to this type of concept, and moreover figures constructed on the surface, their angles, their surface area, their total curvature as well as the connecting of points by curves of shortest length, and similar concepts, all belong to this class.”
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Wells, R.O. (2017). Gauss and Intrinsic Differential Geometry. In: Differential and Complex Geometry: Origins, Abstractions and Embeddings. Springer, Cham. https://doi.org/10.1007/978-3-319-58184-2_4
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DOI: https://doi.org/10.1007/978-3-319-58184-2_4
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