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.

On Kaehler Immersions

1972 ◽  
Vol 24 (6) ◽  
pp. 1178-1182 ◽  
Author(s):  
Koichi Ogiue
Keyword(s):  
Projective Space ◽  
Sectional Curvature ◽  
Complex Projective Space ◽  
Holomorphic Sectional Curvature ◽  
Kaehler Manifold ◽  
Totally Geodesic ◽  
Complex Submanifold ◽  
Complex Projective

Let be an (n + p)-dimensional Kaehler manifold of constant holomorphic sectional curvature . B. O'Neill [3] proved the following result.PROPOSITION A. Let M be an n-dimensional complex submanifold immersed in . If and if the holomorphic sectional curvature of M with respect to the induced Kaehler metric is constant, then M is totally geodesic.He also gave the following example: There is a Kaehler imbedding of an w-dimensional complex projective space of constant holomorphic sectional curvature ½ into an -dimensional complex projective space of constant holomorphic sectional curvature 1. This shows that Proposition A is the best possible.

scholarly journals Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds

2020 ◽  
Author(s):  
Leandro Lichtenfelz ◽  
Gerard Misiołek ◽  
Stephen C Preston
Keyword(s):  
Euler Equations ◽  
Riemannian Geometry ◽  
Exponential Map ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Ideal Fluids ◽  
Volume Preserving ◽  
The Way

Abstract We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples, we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston’s eight model geometries.

scholarly journals THE MINIMAL WITH CONSTANT SECTIONAL CURVATURE IN

2015 ◽  
Vol 99 (1) ◽  
pp. 63-75 ◽  
Author(s):  
SEN HU ◽  
KANG LI
Keyword(s):  
Projective Space ◽  
Sectional Curvature ◽  
Constant Curvature ◽  
Complex Projective Space ◽  
Constant Sectional Curvature ◽  
Complex Projective

It is known that the minimal 3-spheres of CR type with constant sectional curvature have been classified explicitly, and also that the weakly Lagrangian case has been studied. In this paper, we provide some examples of minimal 3-spheres with constant curvature in the complex projective space, which are neither of CR type nor weakly Lagrangian, and give the adapted frame of a minimal 3-sphere of CR type with constant sectional curvature.

scholarly journals WARPED PRODUCT SEMI-SLANT SUBMANIFOLDS IN LOCALLY RIEMANNIAN PRODUCT MANIFOLDS

2008 ◽  
Vol 77 (2) ◽  
pp. 177-186 ◽  
Author(s):  
MEHMET ATÇEKEN
Keyword(s):  
Warped Product ◽  
Slant Submanifold ◽  
Riemannian Product ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Slant Submanifolds ◽  
Geodesic Submanifold

AbstractIn this paper, we prove that there are no warped product proper semi-slant submanifolds such that the spheric submanifold of a warped product is a proper slant. But we show by means of examples the existence of warped product semi-slant submanifolds such that the totally geodesic submanifold of a warped product is a proper slant submanifold in locally Riemannian product manifolds.

scholarly journals Kählerian submanifolds in a complex projective space with second fundamental form of polynomial type

1984 ◽  
Vol 96 ◽  
pp. 61-70
Author(s):  
Ryoichi Takagi
Keyword(s):  
Projective Space ◽  
Sectional Curvature ◽  
Covariant Derivative ◽  
Complex Projective Space ◽  
Second Fundamental Form ◽  
Holomorphic Sectional Curvature ◽  
Fundamental Tensor ◽  
Polynomial Type ◽  
Complex Projective

Let PN be an iV-dimensional complex projective space with Fubini-Study metric of constant holomorphic sectional curvature, and M be a Kählerian submanifold in PN. Let H be the second fundamental tensor + of M, and be the covariant derivative of type (1, 0) on M.

scholarly journals On the index of symmetric spaces

2018 ◽  
Vol 2018 (737) ◽  
pp. 33-48 ◽  
Author(s):  
Jürgen Berndt ◽  
Carlos Olmos
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Riemannian Symmetric Space ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
Riemannian Symmetric Spaces

AbstractLetMbe an irreducible Riemannian symmetric space. The index ofMis the minimal codimension of a (nontrivial) totally geodesic submanifold ofM. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less than or equal to 3.

scholarly journals On totally geodesic submanifolds in the Jacobian locus

2015 ◽  
Vol 26 (01) ◽  
pp. 1550005 ◽  
Author(s):  
Elisabetta Colombo ◽  
Paola Frediani ◽  
Alessandro Ghigi
Keyword(s):  
Upper Bound ◽  
Positive Characteristic ◽  
Second Fundamental Form ◽  
Shimura Varieties ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifold ◽  
The Second Fundamental Form ◽  
Totally Geodesic Submanifolds ◽  
Cyclic Covers

We study submanifolds of Ag that are totally geodesic for the locally symmetric metric and which are contained in the closure of the Jacobian locus but not in its boundary. In the first section we recall a formula for the second fundamental form of the period map Mg ↪ Ag due to Pirola, Tortora and the first author. We show that this result can be stated quite neatly using a line bundle over the product of the curve with itself. We give an upper bound for the dimension of a germ of a totally geodesic submanifold passing through [C] ∈ Mg in terms of the gonality of C. This yields an upper bound for the dimension of a germ of a totally geodesic submanifold contained in the Jacobian locus, which only depends on the genus. We also study the submanifolds of Ag obtained from cyclic covers of ℙ1. These have been studied by various authors. Moonen determined which of them are Shimura varieties using deep results in positive characteristic. Using our methods we show that many of the submanifolds which are not Shimura varieties are not even totally geodesic.

scholarly journals The index conjecture for symmetric spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jürgen Berndt ◽  
Carlos Olmos
Keyword(s):  
Symmetric Spaces ◽  
Systematic Approach ◽  
Riemannian Symmetric Space ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Geodesic Submanifold ◽  
Totally Geodesic Submanifolds ◽  
Higher Rank ◽  
Differential Geom

AbstractIn 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank {\leq 2}, but for higher rank it was unclear how to tackle the problem. In [J. Berndt, S. Console and C. E. Olmos, Submanifolds and holonomy, 2nd ed., Monogr. Res. Notes Math., CRC Press, Boca Raton 2016], [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], [J. Berndt and C. Olmos, The index of compact simple Lie groups, Bull. Lond. Math. Soc. 49 2017, 5, 903–907], [J. Berndt and C. Olmos, On the index of symmetric spaces, J. reine angew. Math. 737 2018, 33–48], [J. Berndt, C. Olmos and J. S. Rodríguez, The index of exceptional symmetric spaces, Rev. Mat. Iberoam., to appear] we developed several approaches to this problem, which allowed us to calculate the index for many symmetric spaces. Our systematic approach led to a conjecture, formulated first in [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], for how to calculate the index. The purpose of this paper is to verify the conjecture.

scholarly journals HERMITIAN SURFACES OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE II

1992 ◽  
Vol 23 (2) ◽  
pp. 137-143
Author(s):  
TAKUJI SATO ◽  
KOUEI SEKIGAWA
Keyword(s):  
Sectional Curvature ◽  
Positive Constant ◽  
Holomorphic Sectional Curvature ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Complex Projective

The present paper ss a continuation of our previous work [7]. We shall prove that a compact Hernutian surface of pointwise positive constant holomorphic sectional curvature is biholomorphica.lly equivalent to a complex projective surface.

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