Classification of conformal minimal immersions of constant curvature from $$S^2$$ S 2 to $$Q_n$$ Q n

2016 ◽  
Vol 196 (3) ◽  
pp. 1001-1023 ◽  
Author(s):  
Xiaoxiang Jiao ◽  
Mingyan Li
Keyword(s):  
Constant Curvature ◽  
Minimal Immersions

scholarly journals Classification of conformal minimal immersions of constant curvature from $S^2$ to $Q_3$

2016 ◽  
Vol 68 (2) ◽  
pp. 863-883 ◽  
Author(s):  
Mingyan LI ◽  
Xiaoxiang JIAO ◽  
Ling HE
Keyword(s):  
Constant Curvature ◽  
Minimal Immersions

Minimal immersions of S2 and ℝP2 into ℂPn with few higher order singularities

1992 ◽  
Vol 111 (1) ◽  
pp. 93-101 ◽  
Author(s):  
J. Bolton ◽  
L. M. Woodward ◽  
L. Vrancken
Keyword(s):  
Sectional Curvature ◽  
Constant Curvature ◽  
Unit Sphere ◽  
Conformal Structure ◽  
Holomorphic Sectional Curvature ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Minimal Immersions ◽  
Complex Projective

In this paper we extend ideas developed in 2, 4 to study certain minimal immersions of S2 and ℝP2 into ℂPn. Here S2 denotes the unit sphere in ℝ3 with its standard conformal structure and ℝP2 is S2 factored out by the antipodal map, while ℂPn denotes complex projective n-space equipped with the FubiniStudy metric of constant holomorphic sectional curvature 4. Since ℝPn with its standard metric of constant curvature 1 is included in ℂPn as a totally geodesic submanifold, this includes the case of minimal immersions into the unit sphere Sn(1) with its standard metric.

scholarly journals CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE

2005 ◽  
Vol 48 (2) ◽  
pp. 337-364 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Constant Curvature ◽  
Euclidean Plane ◽  
Lagrangian Submanifolds ◽  
Real Space ◽  
Point Of View ◽  
Space Forms ◽  
Lagrangian Surfaces ◽  
Real Space Forms ◽  
Geometric Point

AbstractOne of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

scholarly journals A Note on the Characterization of Two-Dimensional Quasi-Einstein Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2013
Author(s):  
Gabriel Bercu
Keyword(s):  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Product Type ◽  
Two Dimensional ◽  
Einstein Manifolds ◽  
Local Coordinates ◽  
Elementary Methods

In this article, we aim to introduce new classes of two-dimensional quasi-Einstein pseudo-Riemannian manifolds with constant curvature. We also give a classification of 2D quasi-Einstein manifolds of warped product type working in local coordinates. All the results are obtained by elementary methods.

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