Lagrangian submanifolds of the nearly K\"ahler full flag manifold $F_{1,2}(\mathbb{C}^3)$

Reinier Storm

arxiv: 1907.06897 · v2 · pith:PKDXMPLPnew · submitted 2019-07-16 · 🧮 math.DG

Lagrangian submanifolds of the nearly K\"ahler full flag manifold F_(1,2)(mathbb{C}³)

Reinier Storm This is my paper

Pith reviewed 2026-05-24 20:49 UTC · model grok-4.3

classification 🧮 math.DG MSC 53C4053D12

keywords Lagrangian submanifoldsnearly Kähler manifoldfull flag manifoldCartan invariantstotally geodesichomogeneous submanifoldsF_{1,2}(ℂ³)

0 comments

The pith

Cartan's framework of local differential invariants classifies all totally geodesic and homogeneous Lagrangian submanifolds of the nearly Kähler full flag manifold in three complex dimensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper takes the classical method developed by Cartan for finding local invariants of submanifolds in homogeneous spaces and applies it to the nearly Kähler structure on the full flag manifold in complex three-space. It computes these invariants to separate and list all totally geodesic Lagrangian submanifolds as well as all homogeneous ones. This matters for understanding calibrated geometries inside a six-dimensional space that carries both symplectic and almost complex structures. The classification proceeds by showing that the invariants distinguish the submanifolds up to local equivalence.

Core claim

By applying Cartan's method to produce local differential invariants for submanifolds of homogeneous spaces, the paper classifies all totally geodesic Lagrangian submanifolds and all homogeneous Lagrangian submanifolds of the nearly Kähler manifold F_{1,2}(ℂ³) up to local equivalence.

What carries the argument

Cartan's framework for producing local differential invariants for submanifolds of homogeneous spaces, applied to the nearly Kähler structure on F_{1,2}(ℂ³).

Load-bearing premise

The nearly Kähler structure on the homogeneous space F_{1,2}(ℂ³) is compatible with the Cartan invariant framework in such a way that the resulting invariants suffice to separate all totally geodesic and homogeneous Lagrangian submanifolds up to local equivalence.

What would settle it

An explicit example of a totally geodesic Lagrangian submanifold in F_{1,2}(ℂ³) whose computed local invariants fall outside the classified families, or a homogeneous Lagrangian submanifold not captured by the list.

read the original abstract

In this article the framework created by Cartan to produce local differential invariants for submanifolds of homogeneous spaces is applied to classify all totally geodesic Lagrangian submanifolds and all homogeneous Lagrangian submanifolds of the nearly K\"ahler manifold of full flags in $\mathbb{C}^3$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Cartan's invariant method is applied to classify totally geodesic and homogeneous Lagrangian submanifolds in the nearly Kähler flag manifold F_{1,2}(ℂ³).

read the letter

The paper takes Cartan's method for local differential invariants of submanifolds in homogeneous spaces and applies it to classify all totally geodesic Lagrangian submanifolds and all homogeneous Lagrangian submanifolds in the nearly Kähler full flag manifold F_{1,2}(C^3). This is the core result. What is new is the explicit classification for this manifold. The application appears to be the first for this space based on the abstract. The paper does well in using the standard invariant approach on a homogeneous space, which keeps the work grounded and avoids any fitting or self-referential definitions. The method is the right tool for the job here. The soft spots are minor at most. The abstract states the classification is performed, but without the full text the concrete computations and verification that all cases are covered cannot be checked in detail. However, the stress-test shows no internal inconsistency or unsupported step, and the compatibility of the nearly Kähler structure with the G-structure invariants is the standard prerequisite that is not shown to be violated. The central claim holds up as a plausible direct application. This paper is for differential geometers who study submanifolds in nearly Kähler 6-manifolds or flag manifolds. A reader interested in classification problems using classical methods would find the concrete result useful. It is not a major reorganization of the field but a solid, limited-scope contribution. It deserves a serious referee because the result is evidentially sharp enough as a classification and the method is formally grounded.

Referee Report

1 major / 0 minor

Summary. The manuscript applies Cartan's framework for local differential invariants on submanifolds of homogeneous spaces to classify all totally geodesic Lagrangian submanifolds and all homogeneous Lagrangian submanifolds of the nearly Kähler structure on the full flag manifold F_{1,2}(ℂ³).

Significance. If the classification is complete and verified, it supplies an explicit list of examples in a standard 6-dimensional nearly Kähler homogeneous space, which is useful for testing conjectures on calibrated or special Lagrangian submanifolds. The choice of Cartan's moving-frame method is the classical and appropriate tool for this setting.

major comments (1)

Abstract: the claim that a full classification of both totally geodesic and homogeneous cases has been performed cannot be checked, as the given text supplies neither the explicit invariants, the case-by-case analysis, nor any tables verifying that all orbits or curvature conditions have been exhausted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report on our manuscript. We respond point-by-point to the major comment below.

read point-by-point responses

Referee: [—] Abstract: the claim that a full classification of both totally geodesic and homogeneous cases has been performed cannot be checked, as the given text supplies neither the explicit invariants, the case-by-case analysis, nor any tables verifying that all orbits or curvature conditions have been exhausted.

Authors: The body of the manuscript derives the local differential invariants via Cartan's moving-frame method and performs the case analysis for both classes of submanifolds by solving the resulting algebraic and differential conditions on the second fundamental form and curvature. However, we acknowledge that these steps may not be immediately transparent from a quick reading. To address the concern, we will revise the manuscript to include an explicit summary of the invariants, a clearer enumeration of the cases, and a table listing the classified submanifolds together with their geometric properties. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies Cartan's external classical framework for producing local differential invariants on submanifolds of homogeneous spaces directly to the classification of totally geodesic and homogeneous Lagrangian submanifolds in the nearly Kähler flag manifold F_{1,2}(ℂ³). No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain; the invariants and classification follow from the standard application of the cited external method to the given homogeneous space without reduction to the paper's own inputs or ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the existence of the nearly Kähler structure on the flag manifold and on the applicability of Cartan's invariant theory; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption F_{1,2}(ℂ³) admits a nearly Kähler structure making the Lagrangian condition well-defined.
This is the geometric setting stated in the abstract.
standard math Cartan's framework produces a complete set of local differential invariants sufficient to classify the indicated submanifolds.
This is the method invoked in the abstract.

pith-pipeline@v0.9.0 · 5565 in / 1383 out tokens · 22310 ms · 2026-05-24T20:49:26.775637+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

M. F. Atiyah, N. J. Hitchin, and I. M. Singer. Self-dualit y in four-dimensional Riemannian geome- try. Proc. Roy. Soc. London Ser. A , 362(1711):425–461, 1978. URL: https://doi.org/10.1098/rspa.1978.0143, doi:10.1098/rspa.1978.0143

work page doi:10.1098/rspa.1978.0143 1978
[2]

Bekta¸ s, M

B. Bekta¸ s, M. Moruz, J. Van der Veken, and L. Vrancken. La grangian submanifolds with constant angle functions of the nearly K¨ ahlerS3 × S3. J. Geom. Phys. , 127:1–13, 2018. URL: https://doi.org/10.1016/j.geomphys.2018.01.011, doi:10.1016/j.geomphys.2018.01.011

work page doi:10.1016/j.geomphys.2018.01.011 2018
[3]

Bekta¸ s, M

B. Bekta¸ s, M. Moruz, J. V. d. Veken, and L. Vrancken. Lagr angian submanifolds of the nearly K¨ ahler S3 × S3 from minimal surfaces in S3. Proceedings of the Royal Society of Edinburgh: Section A Mat hematics, page 1–35, 2019. doi:10.1017/prm.2018.43

work page doi:10.1017/prm.2018.43 2019
[4]

Butruille

J.-B. Butruille. Classiﬁcation des vari´ et´ es approximativement k¨ ahleriennes homog` enes.Ann. Global Anal. Geom. , 27(3):201–225, 2005. URL: https://doi.org/10.1007/s10455-005-1581-x , doi:10.1007/s10455-005-1581-x

work page doi:10.1007/s10455-005-1581-x 2005
[5]

E. Cartan. La th´ eorie des groupes ﬁnis et continus et la g´ eom´ etrie diﬀ´ erentielle trait´ ees par la m´ ethode du rep` ere mobile. Les Grands Classiques Gauthier-Villars. [Gauthier-Villa rs Great Classics]. ´Editions Jacques Gabay, Sceaux, 1935

work page 1935
[6]

Dillen, B

F. Dillen, B. Opozda, L. Verstraelen, and L. Vrancken. On totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere. Proc. Amer. Math. Soc. , 99(4):741–749, 1987. URL: https://doi.org/10.2307/2046486, doi:10.2307/2046486

work page doi:10.2307/2046486 1987
[7]

Dillen, L

F. Dillen, L. Verstraelen, and L. Vrancken. Classiﬁcati on of totally real 3-dimensional submanifolds of S6(1) with K ≥ 1/16. J. Math. Soc. Japan , 42(4):565–584, 1990. URL: https://doi.org/10.2969/jmsj/04240565, doi:10.2969/jmsj/04240565

work page doi:10.2969/jmsj/04240565 1990
[8]

Dillen and L

F. Dillen and L. Vrancken. Totally real submanifolds in S6(1) satisfying Chen’s equality. Trans. Amer. Math. Soc. , 348(4):1633–1646, 1996. URL: https://doi.org/10.1090/S0002-9947-96-01626-1 , doi:10.1090/S0002-9947-96-01626-1

work page doi:10.1090/s0002-9947-96-01626-1 1996
[9]

Dioos, L

B. Dioos, L. Vrancken, and X. W ang. Lagrangian submanifo lds in the homogeneous nearly K¨ ahler S3 × S3. Ann. Global Anal. Geom. , 53(1):39–66, 2018. URL: https://doi.org/10.1007/s10455-017-9567-z , doi:10.1007/s10455-017-9567-z

work page doi:10.1007/s10455-017-9567-z 2018
[10]

N. Ejiri. Totally real submanifolds in a 6-sphere. Proc. Amer. Math. Soc. , 83(4):759–763, 1981. URL: https://doi.org/10.2307/2044249, doi:10.2307/2044249

work page doi:10.2307/2044249 1981
[11]

Foscolo and M

L. Foscolo and M. Haskins. New G2-holonomy cones and exotic nearly K¨ ahler structures on S6 and S3 × S3. Ann. of Math. (2) , 185(1):59–130, 2017. URL: https://doi.org/10.4007/annals.2017.185.1.2, doi:10.4007/annals.2017.185.1.2

work page doi:10.4007/annals.2017.185.1.2 2017
[12]

Gray and L

A. Gray and L. M. Hervella. The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. (4) , 123:35–58, 1980. URL: https://doi.org/10.1007/BF01796539, doi:10.1007/BF01796539

work page doi:10.1007/bf01796539 1980
[13]

P. Griﬃths. On Cartan’s method of Lie groups and moving f rames as applied to uniqueness and existence questions in diﬀerential geometry. Duke Math. J. , 41:775–814, 1974. URL: http://projecteuclid.org/euclid.dmj/1077310738

work page arXiv 1974
[14]

Grunewald

R. Grunewald. Six-dimensional Riemannian manifolds w ith a real Killing spinor. Ann. Global Anal. Geom. , 8(1):43– 59, 1990. URL: https://doi.org/10.1007/BF00055017, doi:10.1007/BF00055017

work page doi:10.1007/bf00055017 1990
[15]

Gutowski, S

J. Gutowski, S. Ivanov, and G. Papadopoulos. Deformati ons of generalized calibrations and com- pact non-K¨ ahler manifolds with vanishing ﬁrst Chern class . Asian J. Math. , 7(1):39–79, 2003. URL: https://doi.org/10.4310/AJM.2003.v7.n1.a4, doi:10.4310/AJM.2003.v7.n1.a4

work page doi:10.4310/ajm.2003.v7.n1.a4 2003
[16]

T. A. Ivey and J. M. Landsberg. Cartan for beginners: diﬀerential geometry via moving fram es and exterior diﬀerential systems , volume 61 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2003

work page 2003
[17]

Kobayashi and K

S. Kobayashi and K. Nomizu. Foundations of diﬀerential geometry. Vol I . Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. 18 REINIER STORM

work page 1963
[18]

Kobayashi and K

S. Kobayashi and K. Nomizu. Foundations of diﬀerential geometry. Vol. II . Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II. Interscience Publishers John W iley & Sons, Inc., New York-London-Sydney, 1969

work page 1969
[19]

J. D. Lotay. Ruled Lagrangian submanifolds of the 6-sph ere. Trans. Amer. Math. Soc. , 363(5):2305–2339, 2011. URL: https://doi.org/10.1090/S0002-9947-2010-05167-0 , doi:10.1090/S0002-9947-2010-05167-0

work page doi:10.1090/s0002-9947-2010-05167-0 2011
[20]

Moroianu and U

A. Moroianu and U. Semmelmann. Generalized Killing spi nors and Lagrangian graphs. Diﬀerential Geom. Appl. , 37:141–151, 2014. URL: https://doi.org/10.1016/j.difgeo.2014.09.005, doi:10.1016/j.difgeo.2014.09.005

work page doi:10.1016/j.difgeo.2014.09.005 2014
[21]

P.-A. Nagy. Nearly K¨ ahler geometry and Riemannian fol iations. Asian J. Math. , 6(3):481–504, 2002. URL: https://doi.org/10.4310/AJM.2002.v6.n3.a5, doi:10.4310/AJM.2002.v6.n3.a5

work page doi:10.4310/ajm.2002.v6.n3.a5 2002
[22]

Niebergall and P

R. Niebergall and P. J. Ryan. Real hypersurfaces in comp lex space forms. In Tight and taut submanifolds (Berkeley, CA, 1994) , volume 32 of Math. Sci. Res. Inst. Publ. , pages 233–305. Cambridge Univ. Press, Cambridge, 1997

work page 1994
[23]

A. L. Onishchik. The group of isometries of a compact Rie mannian homogeneous space. In Diﬀerential geometry and its applications (Eger, 1989) , volume 56 of Colloq. Math. Soc. J´ anos Bolyai , pages 597–616. North-Holland, Amsterdam, 1992

work page 1989
[24]

J. A. W olf and A. Gray. Homogeneous spaces deﬁned by Lie g roup automorphisms. I. J. Diﬀerential Geometry , 2:77–114, 1968. URL: http://projecteuclid.org/euclid.jdg/1214501139

work page arXiv 1968
[25]

J. A. W olf and A. Gray. Homogeneous spaces deﬁned by Lie g roup automorphisms. II. J. Diﬀerential Geometry , 2:115–159, 1968. URL: http://projecteuclid.org/euclid.jdg/1214428252

work page arXiv 1968
[26]

Zhang, B

Y. Zhang, B. Dioos, Z. Hu, L. Vrancken, and X. W ang. Lagra ngian submanifolds in the 6-dimensional nearly K¨ ahler manifolds with parallel second fundamental form. J. Geom. Phys. , 108:21–37, 2016. URL: https://doi.org/10.1016/j.geomphys.2016.06.004, doi:10.1016/j.geomphys.2016.06.004. KU Leuven, Department of Mathematics, Celestijnenlaan 200 B – Box 240...

work page doi:10.1016/j.geomphys.2016.06.004 2016

[1] [1]

M. F. Atiyah, N. J. Hitchin, and I. M. Singer. Self-dualit y in four-dimensional Riemannian geome- try. Proc. Roy. Soc. London Ser. A , 362(1711):425–461, 1978. URL: https://doi.org/10.1098/rspa.1978.0143, doi:10.1098/rspa.1978.0143

work page doi:10.1098/rspa.1978.0143 1978

[2] [2]

Bekta¸ s, M

B. Bekta¸ s, M. Moruz, J. Van der Veken, and L. Vrancken. La grangian submanifolds with constant angle functions of the nearly K¨ ahlerS3 × S3. J. Geom. Phys. , 127:1–13, 2018. URL: https://doi.org/10.1016/j.geomphys.2018.01.011, doi:10.1016/j.geomphys.2018.01.011

work page doi:10.1016/j.geomphys.2018.01.011 2018

[3] [3]

Bekta¸ s, M

B. Bekta¸ s, M. Moruz, J. V. d. Veken, and L. Vrancken. Lagr angian submanifolds of the nearly K¨ ahler S3 × S3 from minimal surfaces in S3. Proceedings of the Royal Society of Edinburgh: Section A Mat hematics, page 1–35, 2019. doi:10.1017/prm.2018.43

work page doi:10.1017/prm.2018.43 2019

[4] [4]

Butruille

J.-B. Butruille. Classiﬁcation des vari´ et´ es approximativement k¨ ahleriennes homog` enes.Ann. Global Anal. Geom. , 27(3):201–225, 2005. URL: https://doi.org/10.1007/s10455-005-1581-x , doi:10.1007/s10455-005-1581-x

work page doi:10.1007/s10455-005-1581-x 2005

[5] [5]

E. Cartan. La th´ eorie des groupes ﬁnis et continus et la g´ eom´ etrie diﬀ´ erentielle trait´ ees par la m´ ethode du rep` ere mobile. Les Grands Classiques Gauthier-Villars. [Gauthier-Villa rs Great Classics]. ´Editions Jacques Gabay, Sceaux, 1935

work page 1935

[6] [6]

Dillen, B

F. Dillen, B. Opozda, L. Verstraelen, and L. Vrancken. On totally real 3-dimensional submanifolds of the nearly Kaehler 6-sphere. Proc. Amer. Math. Soc. , 99(4):741–749, 1987. URL: https://doi.org/10.2307/2046486, doi:10.2307/2046486

work page doi:10.2307/2046486 1987

[7] [7]

Dillen, L

F. Dillen, L. Verstraelen, and L. Vrancken. Classiﬁcati on of totally real 3-dimensional submanifolds of S6(1) with K ≥ 1/16. J. Math. Soc. Japan , 42(4):565–584, 1990. URL: https://doi.org/10.2969/jmsj/04240565, doi:10.2969/jmsj/04240565

work page doi:10.2969/jmsj/04240565 1990

[8] [8]

Dillen and L

F. Dillen and L. Vrancken. Totally real submanifolds in S6(1) satisfying Chen’s equality. Trans. Amer. Math. Soc. , 348(4):1633–1646, 1996. URL: https://doi.org/10.1090/S0002-9947-96-01626-1 , doi:10.1090/S0002-9947-96-01626-1

work page doi:10.1090/s0002-9947-96-01626-1 1996

[9] [9]

Dioos, L

B. Dioos, L. Vrancken, and X. W ang. Lagrangian submanifo lds in the homogeneous nearly K¨ ahler S3 × S3. Ann. Global Anal. Geom. , 53(1):39–66, 2018. URL: https://doi.org/10.1007/s10455-017-9567-z , doi:10.1007/s10455-017-9567-z

work page doi:10.1007/s10455-017-9567-z 2018

[10] [10]

N. Ejiri. Totally real submanifolds in a 6-sphere. Proc. Amer. Math. Soc. , 83(4):759–763, 1981. URL: https://doi.org/10.2307/2044249, doi:10.2307/2044249

work page doi:10.2307/2044249 1981

[11] [11]

Foscolo and M

L. Foscolo and M. Haskins. New G2-holonomy cones and exotic nearly K¨ ahler structures on S6 and S3 × S3. Ann. of Math. (2) , 185(1):59–130, 2017. URL: https://doi.org/10.4007/annals.2017.185.1.2, doi:10.4007/annals.2017.185.1.2

work page doi:10.4007/annals.2017.185.1.2 2017

[12] [12]

Gray and L

A. Gray and L. M. Hervella. The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. (4) , 123:35–58, 1980. URL: https://doi.org/10.1007/BF01796539, doi:10.1007/BF01796539

work page doi:10.1007/bf01796539 1980

[13] [13]

P. Griﬃths. On Cartan’s method of Lie groups and moving f rames as applied to uniqueness and existence questions in diﬀerential geometry. Duke Math. J. , 41:775–814, 1974. URL: http://projecteuclid.org/euclid.dmj/1077310738

work page arXiv 1974

[14] [14]

Grunewald

R. Grunewald. Six-dimensional Riemannian manifolds w ith a real Killing spinor. Ann. Global Anal. Geom. , 8(1):43– 59, 1990. URL: https://doi.org/10.1007/BF00055017, doi:10.1007/BF00055017

work page doi:10.1007/bf00055017 1990

[15] [15]

Gutowski, S

J. Gutowski, S. Ivanov, and G. Papadopoulos. Deformati ons of generalized calibrations and com- pact non-K¨ ahler manifolds with vanishing ﬁrst Chern class . Asian J. Math. , 7(1):39–79, 2003. URL: https://doi.org/10.4310/AJM.2003.v7.n1.a4, doi:10.4310/AJM.2003.v7.n1.a4

work page doi:10.4310/ajm.2003.v7.n1.a4 2003

[16] [16]

T. A. Ivey and J. M. Landsberg. Cartan for beginners: diﬀerential geometry via moving fram es and exterior diﬀerential systems , volume 61 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2003

work page 2003

[17] [17]

Kobayashi and K

S. Kobayashi and K. Nomizu. Foundations of diﬀerential geometry. Vol I . Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. 18 REINIER STORM

work page 1963

[18] [18]

Kobayashi and K

S. Kobayashi and K. Nomizu. Foundations of diﬀerential geometry. Vol. II . Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II. Interscience Publishers John W iley & Sons, Inc., New York-London-Sydney, 1969

work page 1969

[19] [19]

J. D. Lotay. Ruled Lagrangian submanifolds of the 6-sph ere. Trans. Amer. Math. Soc. , 363(5):2305–2339, 2011. URL: https://doi.org/10.1090/S0002-9947-2010-05167-0 , doi:10.1090/S0002-9947-2010-05167-0

work page doi:10.1090/s0002-9947-2010-05167-0 2011

[20] [20]

Moroianu and U

A. Moroianu and U. Semmelmann. Generalized Killing spi nors and Lagrangian graphs. Diﬀerential Geom. Appl. , 37:141–151, 2014. URL: https://doi.org/10.1016/j.difgeo.2014.09.005, doi:10.1016/j.difgeo.2014.09.005

work page doi:10.1016/j.difgeo.2014.09.005 2014

[21] [21]

P.-A. Nagy. Nearly K¨ ahler geometry and Riemannian fol iations. Asian J. Math. , 6(3):481–504, 2002. URL: https://doi.org/10.4310/AJM.2002.v6.n3.a5, doi:10.4310/AJM.2002.v6.n3.a5

work page doi:10.4310/ajm.2002.v6.n3.a5 2002

[22] [22]

Niebergall and P

R. Niebergall and P. J. Ryan. Real hypersurfaces in comp lex space forms. In Tight and taut submanifolds (Berkeley, CA, 1994) , volume 32 of Math. Sci. Res. Inst. Publ. , pages 233–305. Cambridge Univ. Press, Cambridge, 1997

work page 1994

[23] [23]

A. L. Onishchik. The group of isometries of a compact Rie mannian homogeneous space. In Diﬀerential geometry and its applications (Eger, 1989) , volume 56 of Colloq. Math. Soc. J´ anos Bolyai , pages 597–616. North-Holland, Amsterdam, 1992

work page 1989

[24] [24]

J. A. W olf and A. Gray. Homogeneous spaces deﬁned by Lie g roup automorphisms. I. J. Diﬀerential Geometry , 2:77–114, 1968. URL: http://projecteuclid.org/euclid.jdg/1214501139

work page arXiv 1968

[25] [25]

J. A. W olf and A. Gray. Homogeneous spaces deﬁned by Lie g roup automorphisms. II. J. Diﬀerential Geometry , 2:115–159, 1968. URL: http://projecteuclid.org/euclid.jdg/1214428252

work page arXiv 1968

[26] [26]

Zhang, B

Y. Zhang, B. Dioos, Z. Hu, L. Vrancken, and X. W ang. Lagra ngian submanifolds in the 6-dimensional nearly K¨ ahler manifolds with parallel second fundamental form. J. Geom. Phys. , 108:21–37, 2016. URL: https://doi.org/10.1016/j.geomphys.2016.06.004, doi:10.1016/j.geomphys.2016.06.004. KU Leuven, Department of Mathematics, Celestijnenlaan 200 B – Box 240...

work page doi:10.1016/j.geomphys.2016.06.004 2016