Differential Geometry

  • John StillwellEmail author
Part of the Undergraduate Texts in Mathematics book series (UTM)


As mentioned in Chapter 13, calculus made it possible to study nonalgebraic curves: the “mechanical” curves, or transcendental curves as we now call them. Calculus computes not only their basic features, such as tangents and area, but also more sophisticated properties such as curvature. Curvature turns out to be a fundamental concept of geometry, not only for curves, but also for higher-dimensional objects. The concept of curvature is particularly interesting for surfaces, because it can be defined intrinsically. The intrinsic curvature, or Gaussian curvature as it is known, is unaltered by bending the surface, so it can be defined without reference to the surrounding space. This opens the possibility of studying the intrinsic surface geometry. On any smooth surface one can define the distance between any two points (sufficiently close together), and hence “lines” (curves of shortest length), angles, areas, and so on. The question then arises, to what extent does the intrinsic geometry of a curved surface resemble the classical geometry of the plane? For surfaces of constant curvature, the difference is reflected in two of Euclid’s axioms: the axiom that straight lines are infinite, and the parallel axiom. On surfaces of constant positive curvature, such as the sphere, all lines are finite and there are no parallels. On surfaces of zero curvature there may also be finite straight lines; but if all straight lines are infinite the parallel axiom holds. The most interesting case is constant negative curvature, because it leads to a realization of non-Euclidean geometry, as we will see in Chapter 18.


Gaussian Curvature Constant Curvature Geodesic Curvature Logarithmic Spiral Constant Negative Curvature 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bonnet, O. (1848). Mémoire sur la théorie générale des surfaces. J. Éc. Polytech. 19, 1–146.Google Scholar
  2. Dombrowski, P. (1979). 150 Years after Gauss’ “Disquisitiones generales circa superficies curvas”. Paris: Société Mathématique de France. With the original text of Gauss.Google Scholar
  3. Gauss, C. F. (1825). Die Seitenkrümmung. Werke 8: 386–395.Google Scholar
  4. Minding, F. (1839). Wie sich entscheiden lässt, ob zwei gegebene krumme Flächen auf einander abwickelbar sind oder nicht; nebst Bemerkungen über die Flächen von unveränderlichem Krümmungsmasse. J. reine und angew. Math. 19, 370–387.zbMATHGoogle Scholar
  5. Shirley, J. W. (1983). Thomas Harriot: A Biography. New York: The Clarendon Press Oxford University Press.Google Scholar
  6. Stedall, J. (2003). The Greate Invention of Algebra. Oxford: Oxford University Press.zbMATHCrossRefGoogle Scholar
  7. Strubecker, K. (1964). Differentialgeometrie I, II, III. Berlin: Walter de Gruyter.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

Personalised recommendations