Geodesic
Surfaces Given with the Monge Patch in $\mathbb{E}^{4}$
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013), pp. 435-447.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper we consider the surfaces in the Euclidean 4-space $\mathbb{E}^{4}$ given with a Monge patch $z=f(u,v)$, $w=g(u,v)$ and study the curvature properties of these surfaces. We also give some special examples of these surfaces first defined by Yu. Aminov. Finally, we prove that every Aminov surface is a non-trivial Chen surface. 
@article{JMAG_2013_9_a0,
     author = {B. Bulca and K. Arslan},
     title = {Surfaces {Given} with the {Monge} {Patch} in $\mathbb{E}^{4}$},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {435--447},
     publisher = {mathdoc},
     volume = {9},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2013_9_a0/}
}
TY  - JOUR
AU  - B. Bulca
AU  - K. Arslan
TI  - Surfaces Given with the Monge Patch in $\mathbb{E}^{4}$
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2013
SP  - 435
EP  - 447
VL  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JMAG_2013_9_a0/
LA  - en
ID  - JMAG_2013_9_a0
ER  - 
%0 Journal Article
%A B. Bulca
%A K. Arslan
%T Surfaces Given with the Monge Patch in $\mathbb{E}^{4}$
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2013
%P 435-447
%V 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JMAG_2013_9_a0/
%G en
%F JMAG_2013_9_a0
B. Bulca; K. Arslan. Surfaces Given with the Monge Patch in $\mathbb{E}^{4}$. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013), pp. 435-447. http://geodesic.mathdoc.fr/item/JMAG_2013_9_a0/

[1] Yu. Aminov, “Surfaces in $\mathbb{E}^{4}$ with a Gaussian Curvature Coinciding with a Gaussian Torsion up to the Sign”, Math. Notes, 56 (1994), 1211–1215 | DOI | MR | Zbl

[2] Yu. Aminov, The Geometry of Submanifolds, Gordon and Breach Science Publishers, Singapore, 2001 | MR | Zbl

[3] P. J. Besl, R. C. Jain, “Invariant Surface Characteristics for 3D Object Recognition in Range Image”, Comput. Vis. Graph. Image Process., 33 (1986), 33–80 | DOI | Zbl

[4] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs–New York, 1976 | MR | Zbl

[5] B. Y. Chen, Geometry of Submanifols, Dekker, New York, 1973 | MR | Zbl

[6] U. Dursun, “On Product $k$-Chen Type Submanifolds”, Glasgow Math. J., 39 (1997), 243–249 | DOI | MR | Zbl

[7] F. Dillen, L. Verstraelen, L. Vrancken, G. Zafindratafa, “Classification of Polynomial Translation Hypersurfaces of Finite Type”, Results in Math., 27 (1995), 244–249 | DOI | MR | Zbl

[8] H. Gluck, “Higher Curvatures of Curves in Euclidean Space”, Amer. Math. Monthly, 73 (1966), 699–704 | DOI | MR | Zbl

[9] J. Glymph, D. Schelden, C. Ceccato, J. Mussel, H. Schober, “A Parametric Strategy for Free-From Glass Structures Using Quadrilateral Planar Facets”, Automation in Construction, 13 (2004), 187–202 | DOI

[10] F. Geysens, L. Verheyen, L. Verstraelen, “Sur les Surfaces A on les Surfaces de Chen”, C.R. Acad. Sc. Paris, I, 211 (1981)

[11] F. Geysens, L. Verheyen, L. Verstraelen, “Characterization and Examples of Chen Submanifolds”, J. Geom., 20 (1983), 47–62 | DOI | MR

[12] E. Iyigün, K. Arslan, G. Öztürk, “A Characterization of Chen Surfaces in $\mathbb{E}^{4}$”, Bull. Malays. Math. Sci. Soc., 31:2 (2008), 209–215 | MR | Zbl

[13] M. M. Lipschutz, Theory and Problems of Differential Geometry, McGraw-Hill, New York, 1969 | Zbl

[14] H. Liu, “Translation Surfaces with Constant Mean Curvature in 3-Dimensinal Spaces”, J. Geom., 64 (1999), 141–149 | DOI | MR | Zbl

[15] B. Rouxel, “Ruled A-Submanifolds in Euclidean Space $\mathbb{E}^{4}$”, Soochow J. Math., 6 (1980), 117–121 | MR | Zbl

[16] H. F. Scherk, “Bemerkungen ber die kleinste Flche innerhalb gegebener Grenzen”, J. R. Angew. Math., 13 (1835), 185208

[17] L. Verstraelen, J. Walrave, S. Yaprak, “The Minimal Translation Surface in Euclidean Space”, Soochow J. Math., 20 (1994), 77–82 | MR | Zbl

[18] P. Wintgen, “Sur l'inégalité de Chen–Willmore”, C.R. Acad. Sci. Paris, 288 (1979), 993–995 | MR | Zbl