GEOMETRICAL DESCRIPTION OF ONE SURFACE IN ECONOMY

Transcription

1 GOMTRICAL DSCRIPTION OF ON SURFAC IN CONOMY a Kaňkoá Abstract The principal object of this paper is the reglar parametric srface M in R defined by the formla The geometrical description methods we are going to se are based on Cartan s moing frame method and on Weingarten map The stdied map is reglar Let : U R R M U R where U is an open neighborhood of the point q= and p= V M R open neighborhood in R of the point where V is an Key words: Cartan s moing frame method Weingarten map Gassian cratre Main cratre moing frame JL Code: C00 Introdction Let U R is an open neighborhood of a point U and means that the rank of Jacobian matri J A sbset dimensional srface in R and the map : U R is gien by the formla R if for each M : U R is a reglar map which M R is called reglar two there eist an open neighborhood V of V of an open sbset U R onto M V The map Now we hae to constrct the moing frame and orthonormal moing frame which is the base for Cartan method The net method is based on Weingarten mapping Cartan s method The moing frame has the form 554

2 n Symbols and are sed instead of etc Vectors and form the basis of the tangent space T M On T M we can constrct moing frame As the ectors and are tangent ectors of M the n is a normal ector Vector N is the nit normal ector Orthonormal moing frame has the form 0 8 Differential d eqals d d d Differential form is:

3 55 d d d 8 8 d d d d We hae

4 557 The differential form is d d d Frther we try to constrct differential form Differential form is d d Now we will constrct forms and d d d d d d d

5 d d d d d The eterior prodct of forms and is d d d d from which follows d d For control we hae d d d 8 d d 558

6 d d d The reslt is d d The differential of the form d is essential for calclation of the Gass cratre Let s stdy the following eqations: d ω ω ω d d d d d d d d d d d K θ θ d 55

7 50 where K is the Gassian cratre We hae K Weingarten method qation j i ij j i gies j i d which means M T M T We hae We hae system of two eqations for Thanks to Cramer s rle we obtain 8 8 Frther we hae

8 We hae Analogically we obtain

9 5 8 8 D

10 5 Weingarten map can be represented by the matri W W As W and W we obtain: W

11 54 det W As was gien before the Gassian cratre is K Frther we obtain

12 H t r W 7 So the trace of the matri W is H trw Conclsion By the method of moing frame we reached that the reslt of Gassian cratre is K By the method of Weingarten mapping we reached that the reslt of Main cratre is H tr W References Bochner S A new iewpoint in differential geometry Canadian Jornal of mathematics Jornal Canadian de Mathematiqes 5: Breš J Kaňka M Some Conditions for a Srface in Mathematica Bohemica 4 4 s to be a Part of the Sphere Cartan É Oeres complètes-partie I Vol Paris: Gathier-Villars 5 Cartan É Oeres complètes-partie II Vol Paris: Gathier-Villars 5 Cartan É Oeres complètes-partie II Vol Paris: Gathier-Villars 5 Cartan É Oeres complètes-partie III Vol Paris: Gathier-Villars 55 S 55

13 Cartan É Oeres complètes-partie III Vol Paris: Gathier-Villars 55 Kaňka M ample of Basic Strctre qations of Riemannian Manifolds Mnds Symbolics 5 s 57 Chern SS Some new Viewpoints in Differential geometry in the large Blletin of the American mathematical Society 4: 0 Kobayashi S Nomiz K Fondations of Differential Geometry New York: Wiley Interscience Kostant B On differential geometry and homogeneos space Proceedings of the national Academy of sciences of the United States of America 5: 58 Kostant B On differential geometry and homogeneos space Proceedings of the national Academy of sciences of the United States of America 5: Nomiz K Lie Grops and Differential Geometry Tokyo: The Mathematical Society of Japan 5 Rach H A contribtion to differential geometry in the large Annals of Mathematics :8 55 Sternberg S Lectres on Differential Geometry nglewood Cliffs NJ: Prentice-Hall 4 Contact a Kaňkoá Uniersity of conomics Prage W Chrchill Sq Prage Czech Repblic kankoa@secz 5