The Notion of a Riemannian Metric

  • Anatolij T. Fomenko
  • Tosiyasu L. Kinii


In the preceding chapter we associated with each curvilinear coordinate system x = (x 1, ... , x n) on a domain C a smooth functional matrix G(x), and these matrices, at each point, behave as a quadratic form under a change-of-coordinate transformations. The matrices G(x) arose from the formula for the element of length of a curve, and we saw that the length of a curve can be calculated in a curvilinear coordinate system using the G(x). The property of transforming at each point like a quadratic form turns out to be of major importance.


Quadratic Form Bilinear Form Tangent Vector Parametrized Surface Euclidean Plane 
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Copyright information

© Springer Japan 1997

Authors and Affiliations

  • Anatolij T. Fomenko
    • 1
  • Tosiyasu L. Kinii
    • 2
    • 3
  1. 1.Department of Differential Geometry and Application, Faculty of Mathematics and MechanicsMoscow State UniversityMoscowRussia
  2. 2.Laboratory of Digital Art and TechnologyTokyo, 113Japan
  3. 3.Senior Partner of MONOLITH Co. Ltd.Tokyo, 106Japan

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