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Journal articles on the topic 'Gauss Curvature'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 May 2022

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1

Defever, Filip. "Conformally flat hypersurfaces with constant Gauss-Kronecker curvature." Bulletin of the Australian Mathematical Society 61, no. 2 (April 2000): 207–16. http://dx.doi.org/10.1017/s0004972700022218.

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We consider 3-dimensional conformally flat hypersurfaces of E4 with constant Gauss-Kronecker curvature. We prove that those with three different principal curvatures must necessarily have zero Gauss-Kronecker curvature.

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2

WANG, DAN, YAJUN YIN, JIYE WU, and ZHENG ZHONG. "THE INTERACTION POTENTIAL BETWEEN MICRO/NANO CURVED SURFACE BODY WITH NEGATIVE GAUSS CURVATURE AND AN OUTSIDE PARTICLE." Journal of Mechanics in Medicine and Biology 15, no. 06 (December 2015): 1540055. http://dx.doi.org/10.1142/s0219519415400552.

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Based on the negative exponential pair potential ([Formula: see text]), the interaction potential between curved surface body with negative Gauss curvature and an outside particle is proved to be of curvature-based form, i.e., it can be written as a function of curvatures. Idealized numerical experiments are designed to test the accuracy of the curvature-based potential. Compared with the previous results, it is confirmed that the interaction potential between curved surface body and an outside particle has a unified expression of curvatures regardless of the sign of Gauss curvature. Further, propositions below are confirmed: Highly curved surface body may induce driving forces, curvatures and the gradient of curvatures are the essential factors forming the driving forces.

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3

Inoguchi, Jun-ichi, Rushan Ziatdinov, and Kenjiro T. Miura. "A Note on Superspirals of Confluent Type." Mathematics 8, no. 5 (May 11, 2020): 762. http://dx.doi.org/10.3390/math8050762.

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Superspirals include a very broad family of monotonic curvature curves, whose radius of curvature is defined by a completely monotonic Gauss hypergeometric function. They are generalizations of log-aesthetic curves, and other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. In this work, we study superspirals of confluent type via similarity geometry. Through a detailed investigation of the similarity curvatures of superspirals of confluent type, we find a new class of planar curves with monotone curvature in terms of Tricomi confluent hypergeometric function. Moreover, the proposed ideas will be our guide to expanding superspirals.

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4

Morgan, Frank. "WHAT IS... Gauss Curvature?" Notices of the American Mathematical Society 63, no. 02 (February 1, 2016): 144–45. http://dx.doi.org/10.1090/noti1333.

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5

Chow, Bennett, and Dong-Ho Tsai. "Nonhomogeneous Gauss Curvature Flows." Indiana University Mathematics Journal 47, no. 3 (1998): 0. http://dx.doi.org/10.1512/iumj.1998.47.1546.

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6

Cheng, Qing-Ming. "Curvatures of complete hypersurfaces in space forms." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 1 (February 2004): 55–68. http://dx.doi.org/10.1017/s0308210500003073.

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In this paper we investigate three-dimensional complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0). We prove that if the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space E4 and the hyperbolic space H4(c) with vanishing Gauss-Kronecker curvature are also presented. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature is bounded from below. In particular, we classify complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature r satisfies r≥ ⅔c.

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7

Xu, Xingwang, and Paul C. Yang. "Remarks on prescribing Gauss curvature." Transactions of the American Mathematical Society 336, no. 2 (February 1, 1993): 831–40. http://dx.doi.org/10.1090/s0002-9947-1993-1087058-5.

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8

Chou, Kai-Seng, and Weifeng Wo. "On hyperbolic Gauss curvature flows." Journal of Differential Geometry 89, no. 3 (November 2011): 455–85. http://dx.doi.org/10.4310/jdg/1335207375.

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9

Kiessling, Michael, and Sagun Chanillo. "Surfaces with prescribed Gauss curvature." Duke Mathematical Journal 105, no. 2 (November 2000): 309–53. http://dx.doi.org/10.1215/s0012-7094-00-10525-x.

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10

Dursun, Uğur, and Rüya Yeğin. "Hyperbolic submanifolds with finite type hyperbolic Gauss map." International Journal of Mathematics 26, no. 02 (February 2015): 1550014. http://dx.doi.org/10.1142/s0129167x15500147.

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We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

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11

Andrews, Ben. "Motion of hypersurfaces by Gauss curvature." Pacific Journal of Mathematics 195, no. 1 (September 1, 2000): 1–34. http://dx.doi.org/10.2140/pjm.2000.195.1.

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12

SOUSA JR., LUIZ A. M. "Rigidity theorems of Clifford Torus." Anais da Academia Brasileira de Ciências 73, no. 3 (September 2001): 327–32. http://dx.doi.org/10.1590/s0001-37652001000300003.

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Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere Sn + 1. Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has (n - 1) principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus <img src="http:/img/fbpe/aabc/v73n3/03ab.gif" alt="03ab.gif (725 bytes)" align="middle">.

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13

Tanaka, Minoru, and Kei Kondo. "The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces." Nagoya Mathematical Journal 209 (March 2013): 23–34. http://dx.doi.org/10.1017/s0027763000010679.

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AbstractWe construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary if the manifold M is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than 2π. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.

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14

Zhang, Lei. "Vanishing Estimates for Fully Bubbling Solutions of SU (n + 1) Toda Systems at a Singular Source." International Mathematics Research Notices 2020, no. 18 (August 8, 2018): 5774–95. http://dx.doi.org/10.1093/imrn/rny183.

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AbstractFor Gauss curvature equation (or more general Toda systems) defined on 2D spaces, the vanishing rate of certain curvature functions on blowup points is a key estimate for numerous applications. However, if these equations have singular sources, very few vanishing estimates can be found. In this article we consider a Toda system with singular sources defined on a Riemann surface and we prove a very surprising vanishing estimates and a reflection phenomenon for certain functions involving the Gauss curvature.

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15

IZUMIYA, SHYUICHI, DONGHE PEI, and TAKASI SANO. "SINGULARITIES OF HYPERBOLIC GAUSS MAPS." Proceedings of the London Mathematical Society 86, no. 2 (March 2003): 485–512. http://dx.doi.org/10.1112/s0024611502013850.

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In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

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16

Bayard, Pierre, Victor Patty, and Federico Sánchez-Bringas. "On Lorentzian surfaces in ℝ2,2." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 1 (January 6, 2017): 61–88. http://dx.doi.org/10.1017/s0308210516000147.

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We study the second-order invariants of a Lorentzian surface in ℝ2,2, and the curvature hyperbolas associated with its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrinsic proof of the vanishing of the total Gauss and normal curvatures of a compact Lorentzian surface. The Gauss map and the second-order invariants are then used to study the asymptotic directions of a Lorentzian surface and discuss their causal character. We also consider the relation of the asymptotic lines with the mean directionally curved lines. We finally introduce and describe the quasi-umbilic surfaces, and the surfaces whose four classical invariants vanish identically.

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17

TANAKA, AYAKO. "SURFACES IN Sn WITH PRESCRIBED GAUSS MAP AND MEAN CURVATURE VECTOR FIELD." International Journal of Mathematics 17, no. 10 (November 2006): 1127–43. http://dx.doi.org/10.1142/s0129167x06003904.

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We give relations between the Gauss map and the mean curvature vector field of a surface in the Euclidean unit n-sphere Sn. These relations are necessary and sufficient conditions for the existence of a surface in Sn with prescribed Gauss map and mean curvature vector field. We show that such surfaces can be expressed explicitly by using given data.

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18

Xin, Y. L. "Curvature estimates for submanifolds with prescribed Gauss image and mean curvature." Calculus of Variations and Partial Differential Equations 37, no. 3-4 (August 20, 2009): 385–405. http://dx.doi.org/10.1007/s00526-009-0268-8.

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19

Andrews, Ben, and Xuzhong Chen. "Surfaces Moving by Powers of Gauss Curvature." Pure and Applied Mathematics Quarterly 8, no. 4 (2012): 825–34. http://dx.doi.org/10.4310/pamq.2012.v8.n4.a1.

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20

Solanilla, Leonardo, William O. Clavijo, and Yessica P. Velasco. "Swimming in Curved Surfaces and Gauss Curvature." Universitas Scientiarum 23, no. 2 (August 28, 2018): 319–31. http://dx.doi.org/10.11144/javeriana.sc23-2.sics.

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The Cartesian-Newtonian paradigm of mechanics establishes that, within an inertial frame, a body either remains at rest or moves uniformly on a line, unless a force acts externally upon it. This crucial assertion breaks down when the classical concepts of space, time and measurement reveal to be inadequate. If, for example, the space is non-flat, an effective translation might occur from rest in the absence of external applied force. In this paper we examine mathematically the motion of a small object or lizard on an arbitrary curved surface. Particularly, we allow the lizard’s shape to undergo a cyclic deformation due exclusively to internal forces, so that the total linear momentum is conserved. In addition to the fact that the deformation produces a swimming event, we prove –under fairly simplifying assumptions that such a translationis some what directly proportional to the Gauss curvature of the surface at the point where the lizardlies.

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21

Hasanis, T., A. Savas-Halilaj, and T. Vlachos. "Minimal hypersurfaces with zero Gauss-Kronecker curvature." Illinois Journal of Mathematics 49, no. 2 (April 2005): 523–29. http://dx.doi.org/10.1215/ijm/1258138032.

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22

Wang, Mu-Tao. "Gauss Maps of the Mean Curvature Flow." Mathematical Research Letters 10, no. 3 (2003): 287–99. http://dx.doi.org/10.4310/mrl.2003.v10.n3.a2.

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23

Tso, Kaising. "Convex hypersurfaces with prescribed Gauss-Kronecker curvature." Journal of Differential Geometry 34, no. 2 (1991): 389–410. http://dx.doi.org/10.4310/jdg/1214447213.

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24

Guan, Pengfei, and Yan Yan Li. "The Weyl problem with nonnegative Gauss curvature." Journal of Differential Geometry 39, no. 2 (1994): 331–42. http://dx.doi.org/10.4310/jdg/1214454874.

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25

FUJIMOTO, Hirotaka. "On the Gauss curvature of minimal surfaces." Journal of the Mathematical Society of Japan 44, no. 3 (July 1992): 427–39. http://dx.doi.org/10.2969/jmsj/04430427.

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26

Suk-Ho Lee and Jin Keun Seo. "Noise removal with Gauss curvature-driven diffusion." IEEE Transactions on Image Processing 14, no. 7 (July 2005): 904–9. http://dx.doi.org/10.1109/tip.2005.849294.

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27

Chopp, D., L. C. Evans, and H. Ishii. "Waiting time effects for Gauss curvature flows." Indiana University Mathematics Journal 48, no. 1 (1999): 0. http://dx.doi.org/10.1512/iumj.1999.48.1556.

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28

Najafi ivaki, Mohammad. "A note on the Gauss curvature flow." Indiana University Mathematics Journal 65, no. 3 (2016): 743–51. http://dx.doi.org/10.1512/iumj.2016.65.5810.

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29

Hua-kui, Liao. "Surfaces with affine Gauss-Kronecker curvature zero." Archiv der Mathematik 76, no. 2 (February 1, 2001): 149–60. http://dx.doi.org/10.1007/s000130050555.

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30

Xin, Yuanlong. "Mean curvature flow with convex Gauss image." Chinese Annals of Mathematics, Series B 29, no. 2 (February 15, 2008): 121–34. http://dx.doi.org/10.1007/s11401-007-0212-1.

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31

Kim, Lami, Ki-ahm Lee, and Eunjai Rhee. "α-Gauss Curvature flows with flat sides." Journal of Differential Equations 254, no. 3 (February 2013): 1172–92. http://dx.doi.org/10.1016/j.jde.2012.10.012.

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32

Huang, Wung-Hong. "Kaluza-Klein reduction of Gauss-Bonnet curvature." Physics Letters B 203, no. 1-2 (March 1988): 105–8. http://dx.doi.org/10.1016/0370-2693(88)91579-1.

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33

Xin, Y. L. "Mean Curvature Flow with Bounded Gauss Image." Results in Mathematics 59, no. 3-4 (April 2, 2011): 415–36. http://dx.doi.org/10.1007/s00025-011-0112-2.

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34

Milán, F. "Pick invariant and affine Gauss-Kronecker curvature." Geometriae Dedicata 45, no. 1 (January 1993): 41–47. http://dx.doi.org/10.1007/bf01667402.

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35

Andrews, Ben, Pengfei Guan, and Lei Ni. "Flow by powers of the Gauss curvature." Advances in Mathematics 299 (August 2016): 174–201. http://dx.doi.org/10.1016/j.aim.2016.05.008.

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36

Labbi, Mohammed-Larbi. "Manifolds with positive second Gauss–Bonnet curvature." Pacific Journal of Mathematics 227, no. 2 (October 1, 2006): 295–310. http://dx.doi.org/10.2140/pjm.2006.227.295.

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37

Guillen, Nestor, and Jun Kitagawa. "Optimal transport and the Gauss curvature equation." Methods and Applications of Analysis 27, no. 4 (2020): 387–404. http://dx.doi.org/10.4310/maa.2020.v27.n4.a5.

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38

Carretero, Paula, and Ildefonso Castro. "A New Approach to Rotational Weingarten Surfaces." Mathematics 10, no. 4 (February 12, 2022): 578. http://dx.doi.org/10.3390/math10040578.

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Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.

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39

Ding, Wei, and Xin Yu Wang. "Curvature Characters of Revolving Curved Surface and Applications in Math Model of Milling Cutter." Advanced Materials Research 299-300 (July 2011): 988–91. http://dx.doi.org/10.4028/www.scientific.net/amr.299-300.988.

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The curvature characters of revolving curved surface are discussed. A list of the formulas of average curvature and Gauss curvature are made. The math model can be used to provide references to design some milling cutter.

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40

Güler, Erhan, Hasan Hacısalihoğlu, and Young Kim. "The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space." Symmetry 10, no. 9 (September 12, 2018): 398. http://dx.doi.org/10.3390/sym10090398.

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We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

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41

GILKEY, P., J. H. PARK, and K. SEKIGAWA. "UNIVERSAL CURVATURE IDENTITIES III." International Journal of Geometric Methods in Modern Physics 10, no. 06 (April 30, 2013): 1350025. http://dx.doi.org/10.1142/s0219887813500254.

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We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler–Lagrange equations associated to the Chern–Gauss–Bonnet formula and show that they are given solely in terms of curvature and the second fundamental form and do not involve covariant derivatives, thus generalizing a conjecture of Berger to this context.

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42

Duffy, Daniel, and John S. Biggins. "Defective nematogenesis: Gauss curvature in programmable shape-responsive sheets with topological defects." Soft Matter 16, no. 48 (2020): 10935–45. http://dx.doi.org/10.1039/d0sm01192d.

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We deploy the Gauss-Bonnet theorem to calculate the Gauss curvature, both singular and finite, developed by initially flat sheets that are programmed with directional patterns of spontaneous distortion containing topological defects.

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43

TURGAY, NURETTIN CENK. "SOME CLASSIFICATIONS OF LORENTZIAN SURFACES WITH FINITE TYPE GAUSS MAP IN THE MINKOWSKI 4-SPACE." Journal of the Australian Mathematical Society 99, no. 3 (August 13, 2015): 415–27. http://dx.doi.org/10.1017/s1446788715000208.

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In this paper we study the Lorentzian surfaces with finite type Gauss map in the four-dimensional Minkowski space. First, we obtain the complete classification of minimal surfaces with pointwise 1-type Gauss map. Then, we get a classification of Lorentzian surfaces with nonzero constant mean curvature and of finite type Gauss map. We also give some explicit examples.

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44

HINOJOSA, PEDRO A., and GILVANEIDE N. SILVA. "The Gauss Map of Complete Minimal Surfaces with Finite Total Curvature." Anais da Academia Brasileira de Ciências 85, no. 4 (November 10, 2013): 1217–26. http://dx.doi.org/10.1590/0001-3765201376911.

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In this paper we are concerned with the image of the normal Gauss map of a minimal surface immersed in ℝ3 with finite total curvature. We give a different proof of the following theorem of R. Osserman: The normal Gauss map of a minimal surface immersed in ℝ3 with finite total curvature, which is not a plane, omits at most three points of��2 Moreover, under an additional hypothesis on the type of ends, we prove that this number is exactly 2.

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45

Hartig, Marie-Sophie. "Approximation of Gaussian Curvature by the Angular Defect: An Error Analysis." Mathematical and Computational Applications 26, no. 1 (February 9, 2021): 15. http://dx.doi.org/10.3390/mca26010015.

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It is common practice in science and engineering to approximate smooth surfaces and their geometric properties by using triangle meshes with vertices on the surface. Here, we study the approximation of the Gaussian curvature through the Gauss–Bonnet scheme. In this scheme, the Gaussian curvature at a vertex on the surface is approximated by the quotient of the angular defect and the area of the Voronoi region. The Voronoi region is the subset of the mesh that contains all points that are closer to the vertex than to any other vertex. Numerical error analyses suggest that the Gauss–Bonnet scheme always converges with quadratic convergence speed. However, the general validity of this conclusion remains uncertain. We perform an analytical error analysis on the Gauss–Bonnet scheme. Under certain conditions on the mesh, we derive the convergence speed of the Gauss–Bonnet scheme as a function of the maximal distance between the vertices. We show that the conditions are sufficient and necessary for a linear convergence speed. For the special case of locally spherical surfaces, we find a better convergence speed under weaker conditions. Furthermore, our analysis shows that the Gauss–Bonnet scheme, while generally efficient and effective, can give erroneous results in some specific cases.

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46

Liu, Shi Bin, Cheng Wang, Yan Ping Yang, and Xing Yan Liu. "Research on Algorithm of Gauss Curvature on Space Mesh Points of Battlefield Large-Scale Terrain and its Visualization." Applied Mechanics and Materials 543-547 (March 2014): 1803–6. http://dx.doi.org/10.4028/www.scientific.net/amm.543-547.1803.

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This paper discusses the calculation method of Gauss Curvature on space mesh points in simplified level-of-detail (LOD) model technique and how to program with C# language at the platform of .NET. Then, with this algorithm, get the curvature value of the mesh points, delete the center point of the small curvature and triangularize the cavity left so as to realize the simplification of the LOD model. In order to show the result of this algorithm, this paper visualizes the LOD model of the three-dimensional terrain model in full detail with Gauss Curvature Algorithm, puts forward the idea of carrying out error analysis on the algorithm with the geometric property of curvature, and compares the result with that with other algorithms. All the example and the error analysis show that this algorithm is not only of fast speed but also with good effect.

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47

Bracken, Paul. "Quaternionic representation of the moving frame for surfaces in Euclidean three-space and Lax pair." International Journal of Mathematics and Mathematical Sciences 2004, no. 15 (2004): 755–62. http://dx.doi.org/10.1155/s0161171204310392.

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The moving frame and associated Gauss-Codazzi equations for surfaces in three-space are introduced. A quaternionic representation is used to identify the Gauss-Weingarten equation with a particular Lax representation. Several examples are given, such as the case of constant mean curvature.

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48

Li, Hai Zhong. "Gauss curvature of Gaussian image of minimal surfaces." Kodai Mathematical Journal 16, no. 1 (1993): 60–64. http://dx.doi.org/10.2996/kmj/1138039705.

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49

Bertrand, Jérôme. "Prescription of Gauss curvature on compact hyperbolic orbifolds." Discrete & Continuous Dynamical Systems - A 34, no. 4 (2014): 1269–84. http://dx.doi.org/10.3934/dcds.2014.34.1269.

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50

Li, Xiaolong, and Kui Wang. "Nonparametric Hypersurfaces Moving by Powers of Gauss Curvature." Michigan Mathematical Journal 66, no. 4 (November 2017): 675–82. http://dx.doi.org/10.1307/mmj/1508810813.

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