The Shape of Differential Geometry in Geometric Calculus

Abstract

We review the foundations for coordinate-free differential geometry in Geometric Calculus. In particular, we see how both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the Shape Operator. The challenge is to adapt this formalism to Conformal Geometric Algebra for wide application in computer science and engineering.

References

  1. 1.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. The University Press, Cambridge (2003) MATHGoogle Scholar
  2. 2.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann, San Francisco (2007) Google Scholar
  3. 3.
    Hestenes, D.: Space–Time Algebra. Gordon and Breach, New York (1966) MATHGoogle Scholar
  4. 4.
    Hestenes, D.: The design of linear algebra and geometry. Acta Appl. Math. 23, 65–93 (1991) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hestenes, D.: Differential forms in geometric calculus. In: Brackx, F. et al. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 269–285. Kluwer, Dordrecht (1993) Google Scholar
  6. 6.
    Hestenes, D.: Invariant body kinematics: I. Saccadic and compensatory eye movements and II. Reaching and neurogeometry. Neural Netw. 7, 65–88 (1994) CrossRefGoogle Scholar
  7. 7.
    Hestenes, D.: New tools for computational geometry and rejuvenation of screw theory. In: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing for Engineering and Computer Science. Springer, London (2009) Google Scholar
  8. 8.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics, 4th printing 1999. Kluwer, Dordrecht (1984) Google Scholar
  9. 9.
    Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley, New York (1966) MATHGoogle Scholar
  10. 10.
    Hicks, N.: Notes on Differential Geometry. Van Nostrand, New York (1965) MATHGoogle Scholar
  11. 11.
    Lasenby, A., Doran, C., Gull, S.: Gravity, gauge theories and geometric algebra. Philos. Trans. R. Soc. Lond. A 356, 161 (2000) Google Scholar
  12. 12.
    Miller, W.: The geometrodynamic content of the Regge equations as illuminated by the boundary of a boundary principle. Found. Phys. 16(2), 143–169 (1986) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Regge, T.: General relativity without coordinates. Nuovo Cimento 19, 558–571 (1961) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rowley, R.: Finite line of charge. Am. J. Phys. 74, 1120–1125 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Sobczyk, G.: Killing vectors and embedding of exact solutions in general relativity. In: Chisholm, J., Common, A. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 227–244. Reidel, Dordrecht (1986) Google Scholar
  16. 16.
    Sobczyk, G.: Simplicial calculus with geometric algebra. In: Micali, A., Boudet, R., Helmstetter, J. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 227–244. Kluwer, Dordrecht (1992) Google Scholar
  17. 17.
    Struik, D.: Lectures on Classical Differential Geometry. Addison Wesley, Reading (1961) MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Arizona State UniversityTempeUSA

Personalised recommendations