Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지

A NOTE ON E. CARTAN'S METHOD OF EQUIVALENCE AND LOCAL INVARIANTS FOR ISOMETRIC EMBEDDINGS OF RIEMANNIAN MANIFOLDS

  • Han, Chong-Kyu (Department of Mathematics Seoul National University ) ;
  • Yoo, Jae-Nyun (Department of Mathematics Pohang University of Science and Technology)
  • Published : 1997.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

By using the method of equivalence of E. Cartan we calculate the local scalar invariants for Riemannian 2-maniolds. We define also a notion of local invariants for submanifolds in $R^{n + d}, n \geq 2, d \geq 1$, in terms of the symmetry of the local isometric embedding equations of Riemannian n-manifolds into $R^{n + d}$. We show that the local invariants obtained by the Cartan's method are the intrinsic expressions of the local invariants in our sense in the casees of surfaces in $R^3$.

Keywords

References

  1. Differential Geometry L. Auslander
  2. Proc. Symp. Pure Math. v.30 Real hypersurfaces in complex manifolds D. Burns;S. Shnider
  3. J. Math. v.23 Compatibility equationas for isometric embeddings of Riemannian manifolds, Rocky Mt. C. K. Cho;C. K. Han
  4. Comm. Korean Math. Soc. v.8 Complete differential systems for certain isometric immersions of Riemannian manifolds C. K. Cho;C. K. Han;J. N. Yoo
  5. Bull. Amer Math. Soc. v.72 The geometry of G-structures S. S. Chern
  6. Differential Geometric Control Theory, Progress in Math v.27 Differential geometric methods interfacing control theory R. B. Gardner;R. Brockett(ed.);R. Millman(ed.);M. Sussman(ed.)
  7. CBMS-NSF Reg. Conf. Series. v.58 Method of equivalence and its applications R. B. Gardner
  8. Partial Differential Relations M. L. Gromov
  9. Proc. Int. Cong. Math. at Kyoto, Math. Soc. Isometric embeddings of Riemannian manifolds M. Gunter
  10. Math. Preprint Nr. The Theorema Egregium for hypersurfaces and rigidity E. Heil
  11. Duke Math. J. Submitted Infinitesimal symmetries and local invariants of isometric embeddings of Riemannian manifolds C. K. Han;J. N. Yoo
  12. Ann. of Math. Studies v.102 Local isometric embeddings H. Jacobowitz
  13. Amer. Math. Soc. An introduction to CR structures H. Jacobowitz
  14. Preprint CR geometry and Cartan geometry M. Kuranishi
  15. Applications of Lie groups to differential equations P.J. Olver
  16. Preprint An algorithm for computing local invariants of isometric embeddings of Riemannian manifolds into Euclidean spaces J. N. Yoo