The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Bangyen Chen - One of the best experts on this subject based on the ideXlab platform.
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differential geometry of rectifying Submanifolds
arXiv: Differential Geometry, 2016Co-Authors: Bangyen ChenAbstract:A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the notion of rectifying Submanifolds in Euclidean spaces. In particular, we prove that a Euclidean submanifold is rectifying if and only if the tangential component of its position vector field is a concurrent vector field. Moreover, rectifying Submanifolds with arbitrary codimension are completely determined.
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Submanifolds with parallel mean curvature vector in riemannian and indefinite space forms
arXiv: Differential Geometry, 2013Co-Authors: Bangyen ChenAbstract:A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we survey some classical and recent results on Submanifolds with parallel mean curvature vector. Special attention is paid to the classification of space-like and Lorentz surfaces with parallel mean curvature vector in Riemannian and indefinite space forms.
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classification of hamiltonian stationary lagrangian Submanifolds of constant curvature in cp3 with positive relative nullity
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: Bangyen Chen, Oscar J GarayAbstract:Abstract A Lagrangian submanifold in a Kaehler manifold is said to be Hamiltonian-stationary if it is a critical point of the area functional restricted to (compactly supported) Hamiltonian variations. In this paper we classify Hamiltonian-stationary Lagrangian Submanifolds of constant curvature in CP 3 with positive relative nullity. As an immediate by-product, several explicit new families of Hamiltonian-stationary Lagrangian Submanifolds in CP 3 are obtained.
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what can we do with nash s embedding theorem
2004Co-Authors: Bangyen ChenAbstract:According to the celebrated embedding theorem of J. F. Nash, every Riemannian manifold can be isometrically embedded in some Euclidean spaces with sufficiently high codimension. An immediate problem concerning Nash's theorem is the following: Problem: What can we do with Nash's embedding theorem ? In other words, what can we do with arbitrary Euclidean Submanifolds of arbitrary high codimension if no local or global assumption were imposed on the submanifold ? In this survey, we present some general optimal solutions to this and related prob- lems. We will also present many applications of the solutions to the theory of Submanifolds as well as to Riemannian geometry.
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representation of flat lagrangian h umbilical Submanifolds in complex euclidean spaces
Tohoku Mathematical Journal, 1999Co-Authors: Bangyen ChenAbstract:The author proved earlier that, a Lagrangian //-umbilical submanifold in complex Euclidean «-space with n>2 is either a complex extensor, a Lagrangian pseudo-sphere, or a flat Lagrangian //-umbilical submanifold. Explicit descriptions of complex extensors and of Lagrangian pseudo-spheres are given earlier. The purpose of this article is to complete the investigation of Lagrangian //-umbilical Submanifolds in complex Euclidean spaces by establishing the explicit description of flat Lagrangian //-umbilical Submanifolds in complex Euclidean spaces. 1. Statements of theorems. We follow the notation and definitions given in (2). In order to establish the complete classification of Lagrangian //-umbilical Submanifolds in Cn we need to introduce the notion of special Legendre curves as follows. Let z: I^S2n~ι cC"beaunit speed Legendre curve in the unit hypersphere S2n~1 (centered at the origin), i.e., z — z(s) is a unit speed curve in S2n~1 satisfying the condition: (z'(s\ iz(s)) = 0 identically. Since z = z(s) is a spherical unit speed curve, = 0 identically. Hence, z(s), iz(s% z'(s\ iz'(s) are orthonormal vector fields defined along the Legendre curve. Thus, there exist normal vector fields P 3, ...,/>" along the Legendre curve such that (1.1) Φ), ι'Φ), z'(s\ iz\s\ PM iPsis), > Pn(s), iPn(s) form an orthonormal frame field along the Legendre curve. By taking the derivatives of = 0 and of =0, we obtain =0 and = — 1, respectively. Therefore, with respect to an orthonormal frame field chosen above, z" can be expressed as
Hakan Mete Tastan - One of the best experts on this subject based on the ideXlab platform.
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biwarped product Submanifolds of a kahler manifold
arXiv: Differential Geometry, 2016Co-Authors: Hakan Mete TastanAbstract:We study biwarped product Submanifolds which are special cases of multiply warped product Submanifolds in Kahler manifolds. We observe the non-existence of such Submanifolds under some circumstances. We show that there exists a non-trivial biwarped product submanifold of a certain type by giving an illustrate example. We also give a necessary and sufficient condition for such Submanifolds to be locally trivial. Moreover, we establish an inequality for the squared norm of the second fundamental form in terms of the warping functions for such Submanifolds. The equality case is also discussed.
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warped product skew semi invariantSubmanifolds of order 1 of a locallyproduct riemannian manifold
Turkish Journal of Mathematics, 2015Co-Authors: Hakan Mete TastanAbstract:We introduce warped product skew semi-invariant Submanifolds of order $1$ of a locally product Riemannian manifold. We give a necessary and sufficient condition for a skew semi-invariant submanifold of order 1 to be a locally warped product. We also establish an inequality between the warping function and the squared norm of the second fundamental form for such Submanifolds. The equality case is also discussed.
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the geometry of hemi slant Submanifolds of a locally product riemannian manifold
Turkish Journal of Mathematics, 2015Co-Authors: Hakan Mete Tastan, Fatma OzdemirAbstract:In the present paper, we study hemi-slant Submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper hemi-slant submanifold to be a hemi-slant product. We also study these types of Submanifolds with parallel canonical structures. Moreover, we give two characterization theorems for the totally umbilical proper hemi-slant Submanifolds. Finally, we obtain a basic inequality involving Ricci curvature and the squared mean curvature of a hemi-slant submanifold of a certain type of locally product Riemannian manifolds.
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warped product skew semi invariant Submanifolds of order 1 of a locally product riemannian manifold
arXiv: Differential Geometry, 2014Co-Authors: Hakan Mete TastanAbstract:We introduce warped product skew semi-invariant Submanifolds of order $1$ of a locally product Riemannian manifold. We give a necessary and sufficient condition for skew semi-invariant submanifold of order 1 to be a locally warped product. We also prove that the invariant distribution which is involved in the definition of the submanifold is integrable under some restrictions. Moreover, we find an inequality between the warping function and the squared norm of the second fundamental form for such Submanifolds. Equality case is also discussed.
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the geometry of hemi slant Submanifolds of a locally product riemannian manifold
arXiv: Differential Geometry, 2014Co-Authors: Hakan Mete Tastan, Fatma OzdemirAbstract:In the present paper, we study hemi-slant Submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution which is involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper hemi-slant submanifold to be a hemi-slant product. We also study this type Submanifolds with parallel canonical structures. Moreover, we give two characterization theorems for the totally umbilical proper hemi-slant Submanifolds. Finally, we obtain a basic inequality involving Ricci curvature and the squared mean curvature of a hemi-slant submanifold of a certain type locally product Riemannian manifold.
Micah Warren - One of the best experts on this subject based on the ideXlab platform.
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on the regularity of hamiltonian stationary lagrangian Submanifolds
Advances in Mathematics, 2019Co-Authors: Jingyi Chen, Micah WarrenAbstract:Abstract We prove a Morrey-type theorem for Hamiltonian stationary Lagrangian Submanifolds of C n : If a C 1 Lagrangian submanifold is a critical point of the volume functional under Hamiltonian variations, then it must be real analytic. Locally, a Hamiltonian stationary Lagrangian submanifold is determined geometrically by harmonicity of its Lagrangian phase function, or variationally by a nonlinear fourth order elliptic equation of the potential function whose gradient graph defines the Hamiltonian stationary Lagrangian Submanifolds locally. Our result shows that Morrey's theorem for minimal Submanifolds admits a complete fourth order analogue. We establish full regularity and removability of singular sets of capacity zero for weak solutions to the fourth order equation with C 1 , 1 norm below a dimensional constant, and to C 1 , 1 potential functions, under certain convexity conditions, whose Lagrangian phase functions are weakly harmonic.
Jingyi Chen - One of the best experts on this subject based on the ideXlab platform.
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on the regularity of hamiltonian stationary lagrangian Submanifolds
Advances in Mathematics, 2019Co-Authors: Jingyi Chen, Micah WarrenAbstract:Abstract We prove a Morrey-type theorem for Hamiltonian stationary Lagrangian Submanifolds of C n : If a C 1 Lagrangian submanifold is a critical point of the volume functional under Hamiltonian variations, then it must be real analytic. Locally, a Hamiltonian stationary Lagrangian submanifold is determined geometrically by harmonicity of its Lagrangian phase function, or variationally by a nonlinear fourth order elliptic equation of the potential function whose gradient graph defines the Hamiltonian stationary Lagrangian Submanifolds locally. Our result shows that Morrey's theorem for minimal Submanifolds admits a complete fourth order analogue. We establish full regularity and removability of singular sets of capacity zero for weak solutions to the fourth order equation with C 1 , 1 norm below a dimensional constant, and to C 1 , 1 potential functions, under certain convexity conditions, whose Lagrangian phase functions are weakly harmonic.
Heiko Von Der Mosel - One of the best experts on this subject based on the ideXlab platform.
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Compactness and isotopy finiteness for Submanifolds with uniformly bounded geometric curvature energies
Communications in Analysis and Geometry, 2018Co-Authors: Sławomir Kolasiński, Paweł Strzelecki, Heiko Von Der MoselAbstract:We prove isotopy finiteness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined on the class of compact, embedded m-dimensional Lipschitz Submanifolds in R n . That is, there are only finitely many isotopy types of such Submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. Moreover, we establish C 1 -compactness: any sequence of Submanifolds with uniformly bounded en- ergy contains a subsequence converging inC 1 to a limit submanifold with the same energy bound. In addition, we show that all geometric curvature energies under consideration are lower semicontinuous with respect to Hausdor-converge nce, which can be used to min- imise each of these energies within prescribed isotopy classes.
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Compactness and isotopy finiteness for Submanifolds with uniformly bounded geometric curvature energies
arXiv: Differential Geometry, 2015Co-Authors: Sławomir Kolasiński, Paweł Strzelecki, Heiko Von Der MoselAbstract:In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz Submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any sequence of Submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold. This result has two applications. The first one is an isotopy finiteness theorem: there are only finitely many isotopy types of such Submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. The second one is the lower semicontinuity - with respect to Hausdorff-convergence of Submanifolds - of all geometric curvature energies under consideration, which can be used to minimise each of these energies within prescribed isotopy classes.
