The Shape of Differential Geometry in Geometric Calculus

@inproceedings{Hestenes2011TheSO,
  title={The Shape of Differential Geometry in Geometric Calculus},
  author={David Hestenes},
  booktitle={Guide to Geometric Algebra in Practice},
  year={2011}
}
  • D. Hestenes
  • Published in
    Guide to Geometric Algebra in…
    2011
  • Mathematics, Computer Science
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. 
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References

SHOWING 1-10 OF 21 REFERENCES
Differential Forms in Geometric Calculus
Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize thatExpand
Simplicial calculus with Geometric Algebra
We construct geometric calculus on an oriented k-surface embedded in Euclidean space by utilizing the notion of an oriented k-surface as the limit set of a sequence of k-chains. This method providesExpand
New Tools for Computational Geometry and Rejuvenation of Screw Theory
  • D. Hestenes
  • Mathematics, Computer Science
  • Geometric Algebra Computing
  • 2010
TLDR
This paper is a comprehensive introduction to a CGA tool kit and designs for a complete system of powerful tools for the mechanics of linked rigid bodies are presented. Expand
The design of linear algebra and geometry
Conventional formulations of linear algebra do not do justice to the fundamental concepts of meet, join, and duality in projective geometry. This defect is corrected by introducing Clifford algebraExpand
Geometric Algebra for Physicists
Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject withExpand
Killing Vectors and Embedding of Exact Solutions in General Relativity
Two ways in which exact solutions of Einstein’s field equations can be classified are by the existence of preferred vector fields, such as Killing vectors, and by its embedding class in a higherExpand
The geometrodynamic content of the Regge equations as illuminated by the boundary of a boundary principle
In this paper the principle that the boundary of a boundary is identically zero (∂○∂≡0) is applied to a skeleton geometry. It is shown that the left-hand side of the Regge equation may be interpretedExpand
Gravity, gauge theories and geometric algebra
  • A. Lasenby, C. Doran, S. Gull
  • Physics
  • Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
  • 1998
A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independentExpand
Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics
1 / Geometric Algebra.- 1-1. Axioms, Definitions and Identities.- 1-2. Vector Spaces, Pseudoscalars and Projections.- 1-3. Frames and Matrices.- 1-4. Alternating Forms and Determinants.- 1-5.Expand
Geometric algebra for computer science - an object-oriented approach to geometry
TLDR
An introduction to Geometric Algebra that will give a strong grasp of its relationship to linear algebra and its significance for 3D programming of geometry in graphics, vision, and robotics is found. Expand
...
1
2
3
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