moving frames method

{{technical|date=May 2025}}

The equivalence moving frames method was introduced by E. Cartan to solve the equivalence problems on submanifolds under the action of a transformation group. In 1974, P. A. Griffiths has paid to the uniqueness and existence problem on geometric differential equations by using the Cartan method of Lie groups and moving frames.{{Cite journal |last=Griffiths |first=P. A. |date=1974 |title=On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry |journal=Duke Math. J. |volume=41 |issue=4 |pages=775–814|doi=10.1215/S0012-7094-74-04180-5 }} Later on, in the 1990s, Fels and Peter J. Olver have presented the moving co-frame method as a new formulation of the classical Cartan's method for finite-dimensional Lie group actions on manifolds.{{Cite journal |last=M. Fels and P. J. Olver |date=1998 |title=Moving coframes-I: A practical algorithm |journal=Acta Appl. Math. |volume=51 |issue=2 |pages=161–213|doi=10.1023/A:1005878210297 }}{{Cite journal |last=M. Fels and P. J. Olver |date=1999 |title=Moving coframes-II: Regularization and theoretical foundations |journal=Acta Appl. Math. |volume=55 |issue=2 |pages=127–208|doi=10.1023/A:1006195823000 }} In the last two decades, the moving frames method has been developed in the general algorithmic and equivariant framework which gives several new powerful tools for finding and classifying the equivalence and symmetry properties of submanifolds, differential invariants, and their syzygies.{{Cite journal |last=Olver P. J. |date=2003 |title=Moving frames |url=https://www.sciencedirect.com/science/article/pii/S0747717103000920 |journal=J. Symb. Comput. |volume=36 |issue=3 |pages=501–512|doi=10.1016/S0747-7171(03)00092-0 |url-access=subscription }}

Moving frames method in applied on a wide variety of problems, including solving the basic symmetry and equivalence problems of polynomials that form the foundation of classical invariant theory,{{Cite journal |last=Kogan |first=I.A. |date=2001 |title=Inductive construction of moving frames |url=https://www.sciencedirect.com/science/article/pii/S0747717103000920 |journal=Journal of Symbolic Computation |volume=285 |pages=157–170|doi=10.1016/S0747-7171(03)00092-0 |url-access=subscription }} classifying the differential invariants for Lie group actions on functions,{{Cite journal |last=Olver |first=Peter J. |date=2023 |title=Projective invariants of images |url=https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/article/projective-invariants-of-images/F2878CD35F9600DF887583808ACD8189 |journal=European Journal of Applied Mathematics |volume=34 |issue=5 |pages=936–946|doi=10.1017/S0956792522000298 }} analyzing the algebraic structure of differential invariants of PDEs,{{Cite journal |last=G Haghighatdoost, M Bazghandi, F Pashaie |date=2025 |title=Differential Invariants of Coupled Hirota-Satsuma KdV Equations |journal=Kragujevac Journal of Mathematics |volume=49 |issue=5 |pages=793–805 |doi=10.46793/KgJMat2505.793H |doi-access=free }} geometry of curves and surfaces in homogeneous spaces.{{Cite journal |last=Mar´ı Beffa, G., Sanders, J.A., and Wang, J.P |date=2003 |title=Relative and absolute differential invariants for conformal curves |journal=J. Lie Theory |volume=13 |pages=213–245}}