**Hermitian
and Quaternionic Structures on Manifolds and Applications**

CNCS grant PNII-ID-PCE-2011-3-0362 contract no. 33/2011

**Project
description:** The
project refers to hermitian and quaternionic structures on manifolds,
and their connections with the geometry of Frobenius manifolds,
harmonic maps and generalized complex geometry. We use methods from
real and complex differential geometry, twistor theory and
isomonodromic deformations.

**Project
coordinator**

·
Dr.
Liana David,
Scientific Researcher II at Institute
of Mathematics "Simion__
__Stoilow__"
of the Romanian Academy__

**Research
team (in alphabetical order)**

**Intermediate
Evaluation Report: **link

**First
phase of the project (01.01-2012-15.12.2012)**

**Expected
results: **Constructions
of F-manifolds and related structures;

**Scientific
report:** link

**Scientific
papers**

·
D.
V. Alekseevsky, *L.
David*:
A note about invariant SKT-structures and generalized Kahler
structures on flag manifolds, **Proc
Edinburgh Math Soc***,*
vol.
55, no. 3, 2012, pag. 543-549. DOI: 10.1017/S001309151000101

·
*G.
Baditoiu:* Classification
of pseudo-Riemannian submersions with totally geodesic fibers from
pseudo-hyperbolic spaces, * Proc
London Math Soc, *vol.
105, no 6 (2012) pag. 1315-1338. DOI: 10.1112/plms/pds027

·
*G.
Baditoiu,* S.
Ianus, A. M. Pastore: Spectral properties of Riemannian Legendre
foliations, * Bull.
Math. Soc. Sci. Math. Roumanie*,
vol. 56(104), no. 2, 2013, pag. 135-150, link

·
*L.
David:* On
adding a variable to a Frobenbius manifold and generalizations, * Geom
Dedicata*,
vol. 167 (2013), p. 189-214, DOI: 10.1007/s10711-012-9809-y.

· L. David: F-manifolds and eventual identities, Oberwolfach Reports, vol. 9 (2), 2012, p. 1389-1392.

·
S.
Marchiafava, *R.
Pantilie:* Twistor
Theory for co-CR quaternionic manifolds and related structures,
* Israel
J. Math*,
195 (2013) p. 347-371, DOI: 10.1007/s11856-013-0001-3.

·
S.
Marchiafava, *R.
Pantilie: *A
note on CR quaternionic maps, **Adv.
Geom***,
*13
(4) (2013), p. 605-612, DOI: 10.1515/advgeom-2013-0006.

·
*R.
Pantilie:* Generalized
quaternionic manifolds*,
** Ann
Mat Pura Applic*,
193(2014), p. 633-641, DOI:10.1007/s10231-012-0294-6.

**Presentations
as Invited Speakers**

·
Liana
David: Singularity
Theory and Integrable__
____Systems__,
Oberwolfach (Germany), April 2012.

· Liana David: The Central European Mathematical Seminar, Brno (Czech Republic), December 2012.

· Radu Pantilie: Brest University Seminar, Brest (France), June 2012.

**Research
Visits Abroad**

· Gabriel Baditoiu: The Erwin Schrödinger International Institute for Mathematical Physics (Vienna), June (15 days), 2012.

· Liana David: Institut des Hautes Etudes Scientifiques (Paris), 2 months 2012.

· Liana David: Brno University (Czech Republic), November (10 days) 2012.

· Radu Pantilie: Brest University (France), 3 months 2012.

**Second
phase of the project (16.12.2012-15.12.2013)**

**Expected
results **(*according
to AAD 2013*):
1) CDV-structures
and obstructions; 2) Submanifold theory in CDV-manifolds; 3)
Foliations in CR-quaternionic geometry.

**Scientific
report:** link

**Scientific
papers **

·
*L.
David,* I.
A. B. Strachan: Symmetries of F-manifolds with eventual identities
and special families of connections, **Ann.
Sc. Norm. Sup. Pisa***,*
Vol.
XIII (2014), issue 3, p. 641-674, DOI: 10.2422/2036-2145.201205_010

·
*L.
David*,
I. A. B. Strachan: tt* Geometry on the Big Phase Space, * Comm.
Math. Physics*,
vol. 329 (1) (2014), p. 295-323, DOI:10.1007/s00200-014-1964-6

·
*R.
Pantilie:* On
the classification of real vector spaces of a quaternionic vector
space, **Proc.
Edinb. Math Soc****,
**vol.
56, no. 2, pag. 615-622, DOI:10.1017/S0013091513000011.

**Presentations
as Invited Speakers**

· Monica Aprodu: Workshop for Young Researchers in Mathematics, Constanta, Romania, May 2013.

· Gabriel Baditoiu: Workshop for Young Researchers in Mathematics, Constanta, Romania May 2013.

· Radu Pantilie: Aniversary Conference, Faculty of Science – 150 Years, University of Bucharest, Romania, 29 August – 1 September 2013.

•

**Research
Visits Abroad**

· Liana David: Brno University (Czech Republic), February (10 days) 2013.

· Radu Pantilie: The University of Rome “La Sapienza”, September (10 days) 2013.

**Workshop:
**The
director of the project is a co-organizer of The
11-th
International Workshop on Differential Geometry and its Applications
– Ploiesti,
September
19-22, 2013.

**Third
phase of the project (16.12.2013-31.12.2014)**

**Expected
results ***(according
to AAD 2014):* 1)
Riemannian submersions; 2) Differential operators in quaternionic
geometry and generalized geometry; 3) Cartan connections in parabolic
geometry.

**Scientific
report: **link

**Scientific
papers **

·
*Monica
Aprodu, *Marian
Aprodu: Holomorphic vector bundles on Kahler manifolds and totally
geodesic foliations on Euclidian open domains, **Diff.
Geom. Applic, **39
(2015) , p. 10-19,
DOI:10.1016/j.difgeo.2015.01.002,
file,
__arxiv:1409.3701__

·
D.
Alekseevsky, *L.
David*:
Tanaka structures (non-holonomic G-structures) and Cartan
connections, ‘**Conformal
and Complex Geometry’**,
volume in the Honour of P. Gauduchon, Editors: A. Moroianu, L.
Ornea, **J.
Geom. Physics, **vol.
91, p. 88-100, (2015); DOI:10.1016/j.geomphys.2015.01.018;
file,
arxiv:1409.8405

·
*L.
David: *On
cotangent manifolds, complex structures and generalized geometry,
**Ann.
Inst. Fourier**,
vol. 66 (1) (2016), p. 1-28 arxiv:1304.3684

·
*R.
Pantilie*:
On the twistor space of a (co-)CR quaternionic manifold, **New-York
J. Math.** 20
(2014), p. 959-971, http://www.nyjm.albany.edu/j/2014/20-46v.pdf

·
*R.
Pantilie: *On
the quaternionic manifolds whose twistor spaces are Fano manifolds,
**Tohoku
Math. J. **(2)
67 (2015) p. 507-511, arXiv:1411.2225

**Presentations
as Invited Speakers at International Workshops**

·
*Gabriel
Baditoiu:* Real
and Complex Differential Geometry, Bucharest (Romania), 8-12
September 2014

·
*Liana
David: *Real
and Complex Differential Geometry,
Bucharest (Romania), 8-12 September 2014.

·
*Radu
Pantilie:* The
10^{th}
Differential
Geometry Day and The Differential Geometry Workshop,
Lund (Sweden), 15-16 May 2014.

**Foreign
visitors:**

Prof.
Dmitri Alekseevsky from *Institute for Information and Transmission
Problems, Moscow, *visited, as an Invited Professor, the host
institution of the project, IMAR. His travel expenses were supported
by the budget of the project.

**Fourth
phase of the project (01.01.2015-30.12.2015)**

**Expected
results***:
*1)
Totally geodesic foliations on open subsets in Euclidian space; 2)
Riemannian submersions (continuation of the first objective from
2014); 3) Regular F-manifolds; 4) Harmonic morphisms

**Scientific
report: **link

**Scientific
papers **

·
*P.
Baird, R. Pantilie: On Ricci solitons and twistorial harmonic
morphisms*,
**sent
for publication**,
file,
arxiv:1210.4688

·
*L.
David, C. Hertling*:
Regular F-manifolds: initial conditions and Frobenius metrics,
accepted in **Ann.
Sc. Norm. Sup. Pisa**,
file,
arXiv:1411.4553

**Presentations
as Invited Speakers **

·
*Liana
David*,
Geometry
and PDE’s, Timisoara (Romania), 29-30 May 2015.

·
*Radu
Pantilie*,
Harmonic
maps, biharmonic maps, harmonic morphisms and related topics,
Calgari (Italy), 9-13 June 2015.

·
*Liana
David*,
University of Hamburg Research Seminar, Hamburg (Germany), 11
November 2015

**Other
participations at conferences**

·
*Monica
Aprodu * participated
at The 8^{th}
Congress
of Romanian Mathematicians,
Iasi, June 2015 and at Workshop
for Young Researchers in Mathematics, Constanta, May 2015.

**Research
visits abroad **

·
*Gabriel
Baditoiu*,
The
interrelation between mathematical physics, number theory, and
noncommutative geometry, Erwin Schrodinger International
Institute for Mathematical Physics, Vienna (Austria), 1-14
March 2015

**Foreign
visitors:**

Prof.
Ian Strachan from *University of Glasgow (Great
Britain), *visited, as an Invited Professor, the host institution
of the project, IMAR. His travel and local expenses were supported by
the budget of the project.

**Fourth
phase of the project (31.12.2015-30.12.2016)**

**Expected
results***:
*1)
Compatible structures on regular F-manifolds (applications of the
Objective 'Regular F-manifolds' from 2015); 2) The geometry of
stratifolds, 3) Cartan connections in parabolic geometry
(continuation of Objective 3 from 2014)

**Scientific
report: link**

**Scientific
papers **

·
M. A. Aprodu*:
On the exponential map on Riemannian polyhedra*,
**sent
for publication**

·
D. V. Alekseevsky, L.
David:
*Prolongation
of Tanaka structures: an alternative approach,*
accepted
in **Annali
di Matematica Pura ed Applicata, **link

**·**
G.
Baditoiu: *Classification
of homogeneous Einstein metrics on pseudo-hyperbolic spaces***,
sent for publication**

**·**
M.
Benyounes, E. Loubeau, R. Pantilie: *Harmonic
morphisms and moment maps on hyper-Kahler manifolds,*
accepted
in **Manuscripta
Math. **DOI:
10.1007/s00229-016-0894-3

**·**
L.
David C. Hertling:
*Hermitian
metrics on F-manifolds*,
**J.
Geom. Physics, **107
(2016), p. 60-78, link

·
R.
Pantilie*:
Quaternionic-like manifolds and Homogeneous Twistor Spaces*,
accepted in **Proc.
Royal Soc. A. **link

**Presentations
as Invited Speakers **

·
*Gabriel
Baditoiu*,
Workshop
for Young Researchers in Mathematics, Constanta (Romania),
May 2016.

·
*Liana
David*,
Geometric
Structures related to Hermitian and almost-Hermitian manifolds,
Hannover (Germany), May 2016.

**Other
participations at conferences**

·
*Monica
Aprodu * participated
at Workshop
for Young Researchers in Mathematics (Constanta, Romania), May
2016.

· Gabriel Baditoiu participated at The XXIVth International Conference on Integrable Systems and Quantum Symmetries (Prague, Czech Republic) June 2016.

**Research
visits abroad **

*Liana David*visited University of Hamburg (Germany)*Radu Pantilie*visited University of Ferrara (Italy)

**Foreign
visitors:**

Prof.
Claus Hertling (*University of Mannheim, Germany) and Prof.
Andrei Moroianu (CNRS France) *visited the host institution of the
project, IMAR, for collaboration on the topics of the project. They
gave a presentation of their research.