Università del Salento: ESE - Salento University Publishing

ItGli autori riferiscono del ritrovamento di Ostrya carpinifolia Scop. e di Quercus trojana Webb lungo la costa della provincia di Brindisi.EnAbout the presence of Ostrya carpinifolia Scop. and Quercus trojana Webb along the coastal plain of the province of Brindisi-The authors report about the finding of Ostrya  carpinifolia Scop. and Quercus trojana Webb along the coast of the province of Brindisi

Medagli, Pietro

Ruggiero, Livio

Bianco, Pasqua

D'Emerico, Saverio

Università del Salento - Coordinamento SIBA

P. MEDAGLI**+, L. RUGGIERO*+, P. BIANCO**, S. D’EMERICO** 
PRESENZA DI OSTRYA CARPINIFOLIA SCOP. 
E DI QUERCUS TROJANA WEBB 
LUNG0 LA PIANA COSTIERA BRINDISINA 
Ostrya carpinifolia Scop. (Carpino nero, Carpinella) 5, secondo 
PIGNATTI (1982), entita tipica dei consorzi di degradazione del bosco 
xerofilo submediterraneo e in Italia risulta assente, di norma, nelle piane 
alluvionali e nella fascia mediterranea sempreverde. La presenza di 
questa specie nella Puglia centro-meridionale, peraltro molto sporadica, 
era fino ad ora nota per alcune vallette fresche e profonde dell’area 
murgiana, come ne1 complesso boschivo delle Pianelle a Martina Franca 
(Ta), posto a1 margine della piattaforma calcarea delle Murge di Sud-Est, 
(MARINOSCI, 1870; BIANCO, 1976) o sul fondo di alcune gravine dell’arco 
jonico tarantino, come in quelle di Laterza (BIANCO, 1976) e di Ginosa 
(PARENZAN, in verbo). Appare, quindi, quantomeno insolito il suo ritrova- 
mento lungo la piana costiera brindisina, in due piccoli nuclei distinti 
collocati in due depressioni all’interno di due complessi boschivi noti con 
i nomi di (<BOSCO del Compare,, e (~BOSCO di Cerano>>. Si tratta di due boschi 
misti con prevalenza di Leccio (Quercus ilex L.) e Roverella (Quercus 
pubescens’willd.). I1 primo bosco sorge a circa 6-7 km a Nord-Ovest dalla 
ci t ta  di Brindisi, mentre il secondo a circa 10 km a Sud-Est dalla stessa. 
E’ evidente che la presenza di questa specie 5 resa possibile dalle 
particolari condizioni microclimatiche che si instaurano nelle depressio- 
ni e che determinano un fenomeno di inversione delle fasce di vegetazio- 
ne. In entrambe le depressioni i3 stato misurato un PH basico del terreno 
e un contenuto di sabbia intorno a1 70%. 
Quercus trojana Webb (Fragno) i3 entita a diffusione mediterraneo- 
orientale che ha  in Puglia e ne1 Materano l’estremo lembo occidentale del 
** Istituto di Botanica dcll’universith, Via Amendola 173 - 70126 Bari. 
* Facolth di Scienze dell’Universith, c.p. 193 - 73100 Lecce. 
+ Gruppo Naturalisti Salentini, Lecce. 
69 
suo areale che risulta circoscritto a un limitato settore dell’area delle 
Murge, secondo la distribuzione di MASELLI (1940) recentemente amplia- 
ta da VITA et al. (1988 e 1989). In questa nota si segnala il ritrovamento 
di una stazione di Quercus trojana Webb ne1 ~BOSCO del Compare,>. La 
stazione 5 costituita da un nucleo di una cinquantina di esemplari arborei 
e arbustivi, con abbondante fruttificazione e con rinnovazione sponta- 
nea. Peril momento risulta prematuro esprimere un parere in merito al- 
l’origine spontanea del ritrovamento che insolitamente si colloca lungo la 
piana costiera brindisina e a1 di fuori dell’areale noto della specie (Fig. 1). 
RIASSUNTO 
Gli autori riferiscono del ritrovamento di Ostrya carpinifolia Scop. 
e di Quercus trojana Webb lungo la costa della provincia di Brindisi. 
Markina 
o Ma.xfurrj 
G O L F 0  
r n ~ n N r o  
Fig. 1 - Distribuzione di Quercus trojana Webb in Italia secondo MASELI,I (1940) 
modificato da VITA et al. (1988 e 1989). I1 tondino indicato dalla freccia 
individua la  posizione del <<BOSCO del Compare,,. 
70 
SUMMARY 
About the presence of Ostrya carpinifolia Scop. and Quercus trojana 
Webb along the coastal plain of the province of Brindisi - The authors 
report about the finding of Ostrya carpinifolia Scop. and Quercus trojana 
Webb along the coast of the province of Brindisi. 
BIBLIOGRAFIA 
BIANCO P., 1976 - Resoconto sulla escursione sociale in Puglia (27-30 aprile 1975) 
della Societj Botanica Italiana. Annali della Fac. di Agraria dell'univ. di 
Bari, 28 : 271 - 317. 
MARINOSCI M. 1870 - La Flora Salentina, 2 : 202, Ed. Salentina, Lecce. 
MASELLI V. G., 1940 - Contributo alla conoscenza delle Querce d'Italia. I1 Fragno, 
PIGNATTI S., 1982 - Flora d'Italia, 1 : 111, Edagricole, Bologna. 
VITA F., CASTELLANETA A. M., FORTE L., DEFILIPPIS R., 1988 - La vegetazione 
spontanea del Lamone di Rutigliano. Umanesimo della Pietra - Verde, 3 : 7 
- 12. 
VITA F., FORTE L., DEFILIPPIS R., 1989 - L'areale del Fragno si estende ne1 bosco 
di Lama Giotta. Umanesimo della Pietra - Verde, 4 : 41 - 44. 
Riv. Forest. Ital., 2 : 20 - 35. 
71 


Italian

Presenza di Ostrya carpinifolia Scop. e di Quercus trojana Webb lungo la piana costiera brindisina

Basset, Alberto

Transitional Waters Bulletin 
TWB, Transit. Waters Bull. 3(2007), 1-2 
ISSN 1825-229X, DOI 10.1285/i1825229Xv1n3p1 
http://siba2.unile.it/ese/twb 
 
 
© 2007 University of Salento - SIBA  http://siba2.unile.it/ese 
 
RESEARCH ARTICLE 
TWReferenceNET: management and sustainable 
development of protected transitional waters.  
Alberto Basset  
 
Di.S.Te.B.A., University of Salento, SP Lecce Monteroni, 73100 Lecce, Italy 
 
 
 
TWReferenceNET has been one of the largest 
European projects dealing with monitoring and 
conservation of transitional water ecosystems 
 
The project involved 22 partners of five 
different countries, Albania, Bulgaria, Greece, 
Italy and Romania, which developed common 
field research activities on 16 lagoons in the 
Eastern Mediterranean and Black Sea, including 
a number of protected ecosystems and Ramsar 
sites (Le Cesine, Torre Guaceto and Grado-
Valle Cavanata in Italy, Karavasta and Patok in 
Albania, Agiasma in Greece, Sinoe and Lehaova 
in Romania), a number of saltworks (Margherita 
di Savoia in Italy, Narta in Albania, Kalloni in 
Greece) as well as a few ecosystems heavily 
impacted by industrial activities (Pialassa-
Baiona and Grado-Marano in Italy, Varna in 
Bulgaria) (Fig. 1). 
 
The  project had three major strategic 
objectives: (1) to ameliorate conservation of 
transitional water natural heritage health in the 
CADSES area, by deepening knowledge, 
producing technological innovation, reinforcing 
regulation and overcoming fragmentation of 
expertise and activities, of scientific, socio-
economic and decision-maker competencies; (2) 
to reinforce territorial integration in the 
CADSES area on the environmental policy 
conservation strategies, according to the UE 
Directives (e.g., Nature 2000, Water Framework 
Directives); (3) to encourage socio-economic 
development in NMS coastal areas, by 
translating the efforts on conservation into 
sustainable development strategies,  creation of 
new jobs opportunities and improvement of 
quality of life. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Common field research activities addressed 
topic within the first two project objectives, 
representing the larges compelling 
intercalibration exercise currently existing at the 
European scale on transitional water 
ecosystems. In fact, one of the project work-
packages was designed in order to support the 
implementation of the Water Framework 
Directive in the Mediterranean and Black Sea 
Eco-Regions. It dealt with trans-national studies 
and innovative strategies on monitoring and 
conservation of protected transitional waters 
and its major objective was to develop 
Fig. 1 – Map of the transitional aquatic ecosystems 
studied in TWReferenceNET. 
TWB 3 (2007) Basset A. 
 
 
© 2007 University of Salento - SIBA  http://siba2.unile.it/ese  2 
methodological tools and standardise procedures 
to improve conservation strategies of public 
administration through  an effective monitoring 
of the protected ecosystem ecological status. 
 
TWReferenceNET fields studies focused on 
three biological quality elements, among those 
reported in the Water Framework Directive with 
respect to transitional water ecosystems, i.e., 
phytoplankton, phytobenthos and benthic 
macro-invertebrates. For every quality element, 
population and community level variability 
were accounted for at a hierarchy of different 
scales:  the within ecosystem level, according to 
spatial differences in the major abiotic niche 
axes (e.g., water salinity and oxygen content, 
nutrients, substrate), the ecosystem level, 
according to differences in the hydrological, 
physiographic and physical-chemical 
component defining the abiotic niche space 
characterizing each ecosystem, and  the eco-
regional scale, according to climate and macro-
distribution of species. Taking advantage of the 
experimental design, comparing a number of 
potential reference condition ecosystems with a 
number of differently stressed ecosystems, with 
which gradients of abiotic pressures were 
considered, the most compelling descriptors of 
ecological status and new descriptors, based on 
body size measurements and conceptually 
founded on the metabolic theory, were 
determined. 
 
The size based approach offers an alternative  
and sensitive technique for describing and 
comparing benthic and pelagic communities. 
Body size is the first intrinsic attribute of each 
organism and numerous physiological and 
ecological processes are linked to it  through  
the same general form 
 
Y = a W b 
 
Where, Y = dependent variable, W = body size, 
a = normalization constant, b = scaling 
exponent. This equation is a power-law 
relationship because dependent variable can be 
expressed as power of another.  The relationship 
is also called scaling law  given that it describes 
how a dependent variable (Y) scales  with a 
indipendent variable (W). On the basis of this 
fundamental relation, population and 
community level parameters scale with body 
size at some expected rate, which are likely to 
represent model reference condition. 
 
In this issue, a number of contributions on both 
structural and functional characteristics of 
transitional water ecosystems, in the light of the 
implementation of the Water Framework 
Directive, are presented. The contributions were 
originally presented to the final meeting of 
TWReferenceNET, which was held in Lecce on 
November 30th – December 2nd, 2007, in the 
section dealing  with trans-national studies and 
innovative strategies on monitoring and 
conservation of protected transitional waters. 
These papers, in a very synthetic form presents 
the bulk of data collected within 
TWReferenceNET and may represent a useful 
literature for researchers working in the area of 
interest regarding the evaluation of ecological 
status in Mediterranean and Black Sea 
transitional water ecosystems.  
 
All manuscripts reported in the issue are funded 
by the INTERREG III B CADSES grant to 
TWReferenceNET.  
 


English

TWRefernceNET: management and sustainable development of protected transitional waters

The main objectives of the present study are:
1-to assess the ecological status of Varna lagoon
2-to identify the environmental factors governing the community pattern

Trayanova, Antoaneta

Moncheva, Snejana

Doncheva, Valentina

Transitional Waters Bulletin 
TWB, Transit. Waters Bull. 3(2007), 33-36 
ISSN 1825-229X, DOI 10.1285/i1825229Xv1n3p33 
http://siba2.unile.it/ese/twb 
 
 
© 2007 University of Salento - SIBA  http://siba2.unile.it/ese 
 
RESEARCH ARTICLE 
Macrozoobenthic communities as a tool for assessment 
the ecological status of Varna lagoon.  
Antoaneta Trayanova, Snejana Moncheva, Valentina Doncheva  
 
Institute of Oceanology – Varna, Bulgarian Academy of Sciences 
 
corresponding author: A. Trayanova e-mail: trayanova@io-bas.bg  
 
 
Abstract 
The main objectives of the present study are: 
1  - to assess the ecological status of Varna lagoon 
2  - to identify the environmental factors governing the community pattern. 
 
 
Introduction 
Benthic invertebrates are used extensively as 
indicator of environmental status because 
numerous studies have demonstrated that 
benthos responds predictably to many kinds of 
natural and anthropogenic stress (Dauer, 1993; 
Tapp et al. 1993; Wilson and Jeffrey 1994; Ahn 
et al., 1995). A number of biotic indices based 
on benthic community health have been 
developed to classify the ecological status of 
coastal and transitional (Borja et al., 2000; 
Muxika et al., 2006; Simboura & Zenetos, 
2002) in accordance with the requirements of 
Water Framework Directive 2000/60/EC. The 
body-size distribution is alternative method for 
studying the structure and disturbance of 
benthic communities. Macroinvertebrates can be 
used to understand the combination of certain 
environmental factors affecting the structure of 
benthic communities. 
 
Material and methods 
The study area is located at the western part of 
the Black Sea coast and includes the Varna 
Lake, the navigation canal between the lake and 
Varna Bay and Varna Bay. Five sampling sites 
(Fig. 1) have been selected taking into account 
the environmental gradient of nutrients, which 
concentration decrease from lake to bay. 
Seasonal samples from water column and soft 
bottom sediments have been collected in late 
autumn of 2004, spring and summer of 2005 to 
study the abiotic parameters of the environment 
and biotic features of macrozoobenthos. In the 
laboratory macroinvertebrates were removed 
from the remaining sediment and separated to 
morphological functional group level. Each 
invertebrate was identified at the lowest 
possible taxonomic level and measured along its 
main axis with image analysis system (Leica 
Mz12, photo-camera Leica IC A, software Leica 
Qwin). For determination of abundance 
individuals of each species or higher taxa were 
counted. Each measured macroinvertebrate was 
desiccated in a stove for at least 72h at 60°C 
and after drying was weighed on an electronic 
micro-scales (± 0.001mg) to determine the 
biomass. Primer 5.0 and AMBI 4.0 Software 
Programme Packages were employed in the 
analysis of data. 
The classification of ecological status, 
following the requirements of WFD 
60/2000/EC, is done by Shannon diversity index 
(H’), a Marine Biotic Index (AMBI) and 
multivariate AMBI (M-AMBI). 
 
Results and discussions  
The biotic indices (species richness S, 
Margaleff index d and Shannon diversity index 
TWB 3 (2007) Trayanova et al. 
 
 
© 2007 University of Salento - SIBA  http://siba2.unile.it/ese  34 
H’), calculated on the abundance data, increase 
in conformity with the diminishing of nutrients 
load thus reflecting the improvement of water 
quality. 
The share of opportunistic species (polychaetes 
and oligochaetes) in the total abundance 
decrease with lowering the concentration of 
nutrients and the share of conservative 
(molluscs) and sensitive (crustaceans) species 
increase in the same direction (Fig. 1). These 
changes of benthic community, i. e. elimination 
of conservative and sensitive species in favour 
of tolerant opportunistic species (Pearson & 
Rosenberg, 1978), reflect the effect of 
environmental conditions on community pattern. 
. 
 
 
Figure 1. Location of sampling stations, values of biotic indices species richness, Margaleff, Shannon diversity 
and percentage share of main taxonomic group in the community structure. 
 
 
The classification of ecological status according 
to value of Shannon diversity index ranks the 
lake site in bad, the canal sites in poor and 
moderate and the bay sites in moderate and poor 
status respectively. The ecological status 
assessed by the index AMBI (Borja et al., 2000) 
classifies the lake site in bad status, the canal 
sites in good and poor, and the bay sites in good 
status. The multivariate AMBI (M-AMBI) 
(Muxika et al., 2006) combining the values of 
AMBI, species richness and Shannon diversity 
categorizes the lake site in bad status, the canal 
sites in bad and poor status and the bay sites in 
moderate and poor ecological status (Tab. 1). 
It is obvious form the table that the overall 
classification of ecological status differentiates 
depending on the indices applied. The exception 
is the lake station which ecological status is 
classified as bad by the three metrics due to the 
dominance of opportunistic species. For the 
second canal site the status falls into two 
neighbour categories. For the rest of the sites 
higher degree of inconsistency in classification 
is observed. 
The size structure of macrozoobenthos reveals 
diversification of body length classes and 
increasing the width of body size distribution 
from lake to bay. The weight per length ratio 
also increases in the same direction suggesting 
that lighter individuals develop in lake 
environment. 
The best combination of variables explaining 
the biological pattern is found to be the salinity, 
the organic carbon percentage in sediments and 
TWB 3 (2007) Macrozoob. comm. as a tool for ass. the ecol. status of Varna lag. 
 
 
© 2007 University of Salento - SIBA  http://siba2.unile.it/ese  35 
the percentage share of sand fraction (Spearman 
rank correlation ? = 0.818). The environmental 
variables and community pattern show good 
correspondence (Fig. 2). 
 
Table 1. Ecological status classification according to the biotic indices  
Site H' AMBI M-AMBI 
Lake Bad Bad Bad 
Canal Poor Good Bad 
Canal Moderate Poor Poor 
Bay Moderate Good Moderate 
Bay Poor Good Poor 
  
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
Figure 2. PCA of selected environmental variables (A) and BIOENV analysis of macrozoobenthos community 
pattern (B). 
 
 
 
Conclusions 
The biotic and multivariate indices for 
evaluation the disturbance of benthic 
communities are reliable tool for categorization 
of the ecological status in accordance with the 
requirements of WFD 60/2000/EC. Because 
some inconsistencies in the classification have 
been detected, the metrics require further 
validation for better understanding of the 
community response to different natural and/or 
anthropogenic pressures. 
Body size distribution of benthic assemblages 
appears to be an objective method for 
assessment of the ecological status. 
The environmental factors affecting the 
structure of benthic community should be taken 
into consideration in the process of ecological 
status scaling. 
 
Acknowledgments 
This paper was funded by an INTERREG IIIB 
CADSES grant to the project 
TWReferenceNET. 
 
References 
Ahn, I. Y., Y. C. Kang and J. W. Choi, 1995. The 
influence of industrial effluents on intertidal 
benthic communities in Panweol, Kyeonggi Bay 
TWB 3 (2007) Trayanova et al. 
 
 
© 2007 University of Salento - SIBA  http://siba2.unile.it/ese  36 
(Yellow Sea) on the west coast of Korea. Marine 
Pollution Bullettin 30:200-206. 
Borja, A., Franco, J., Pérez, V., 2000. A Marine 
Biotic Index to establish the ecological quality of 
soft-bottom benthos within European estuarine and 
coastal environments. Marine Pollution Bulletin 
40: 1100–1114. 
Dauer, D. M., 1993. Biological criteria, 
environmental health and estuarine macrobenthic 
community structure. Marine Pollution Bulletin 
26: 249–257. 
Muxika, I., Borja A., Bald J., 2006. Using historical 
data, expert judgement and multivariate analysis in 
assessing reference conditions and benthic 
ecological status, according to the European Water 
Framework Directive. Marine Pollution Bulletin 
(in press). 
Pearson, T. H., Rosenberg R., 1978. Macrobenthic 
succession in relation to organic enrichment and 
pollution of the marine environment. 
Oceanography and Marine Biology Annual Review 
5: 229–311. 
Simboura, N. and A. Zenetos, 2002. Benthic 
indicators to use in Ecological Quality 
Classification of Mediterranean soft bottom marine 
ecosystems, including a new Biotic Index. 
Mediterranean Marine Science 3/2: 77-111. 
Tapp, J. F., N. Shillabeer and C. M. Ashman., 1993. 
Continued observation of the benthic fauna of the 
industrialised Tees estuary, 1979-1990. Journal of 
Experimental Make Biology and Ecology 172:67-
80. 
Wilson, J. G. and D. W. Jeffrey. 1994. Benthic 
biological pollution indices in estuaries, p. 31l-
327. In J. M. Kramer (ed.), Biomonitoring of 
Coastal Waters and Estuaries. CRC Press, Boca 
Raton, Florida. 
 


Macrozoobenthic communities as a tool for assessment the ecological status of Varna lagoon

We classify algebraic curvature tensors such that the Ricci operator &#961; is simple (i.e. &#961;is complex diagonalizable and Spec{&#961;}={a} or Spec{&#961;}={a1&#x00b1;a2&#x221A;-1}) and which are Jacobi–Ricci commuting (i.e. &#961;J(v)=J(v)&#961; for all v)

Gilkey, P.

Nikcevic, S.

Note di Matematica 28, suppl. n. 1, 2009, 341–348.
The classification of simple Jacobi–Ricci
commuting algebraic curvature tensors
P. Gilkey
Mathematics Department, University of Oregon,
Eugene Oregon 97403 USA
gilkey@uoregon.edu
S. Nikcˇevic´
Mathematical Institute, SANU,
Knez Mihailova 35, p.p. 367, 11001 Belgrade, Serbia
stanan@mi.sanu.ac.yu
Received: 10/10/07; accepted: 30/10/07.
Abstract. We classify algebraic curvature tensors such that the Ricci operator ρ is simple
(i.e. ρ is complex diagonalizable and Spec{ρ} = {a} or Spec{ρ} = {a1 ± a2
√−1} ) and which
are Jacobi–Ricci commuting (i.e. ρJ(v) = J(v)ρ for all v).
Keywords: algebraic curvature tensor, Jacobi operator, Ricci operator
MSC 2000 classification: primary 53C20
This is dedicated to Professor Oldrich Kowalski
The study of curvature is fundamental in differential geometry. It is often
convenient to work first in an abstract algebraic context and then subsequently
to pass to the geometrical setting. We say that M := (V, 〈·, ·〉, A) is a model if
〈·, ·〉 is a non-degenerate inner-product of signature (p, q) on a real vector space
V of dimension m = p+ q and if A ∈ ⊗4V ∗ is an algebraic curvature tensor, i.e.
a 4-tensor which has the symmetries of the Riemann curvature tensor:
A(v1, v2, v3, v4) = −A(v2, v1, v3, v4) = A(v3, v4, v1, v2),
A(v1, v2, v3, v4) +A(v2, v3, v1, v4) +A(v3, v1, v2, v4) = 0 .
(1)
If P is a point of a pseudo-Riemannian manifold M = (M,g), then the associ-
ated model is defined by setting M(M, P ) := (TPM,gP , RP ) where RP is the
curvature tensor of the Levi–Civita connection; every model is geometrically
realizable in this fashion. Consequently the study of algebraic curvature tensors
plays a central role in many geometric investigations.
If M is a model, then Jacobi operator J, the skew-symmetric curvature op-
erator R, and the Ricci operator ρ are defined by the identities:
Note di Matematica
Note Mat. 1(2008), suppl. n. 1, 341-348
ISSN 1123-2536, e-ISSN 1590-0932
DOI 10.1285/i15900932v28n1supplp341
http://siba2.unile.it/notemat
_____________________________________________________________________________________
342 P. Gilkey, S. Nikcˇevic´
〈J(x)y, z〉 = A(y, x, x, z),
〈R(x, y)z,w〉 = A(x, y, z, w),
〈ρx, y〉 = Tr{z → 12R(z, x)y + 12R(z, y)x} .
One says M is Einstein if ρ = a id; a is called the Einstein constant.
The study of commutativity properties of natural operators defined by the
curvature tensor was initiated by Stanilov [4, 5] and has proved to be a very
fruitful one; we refer to [1] for a survey of the field and for a more complete
bibliography than is possible to present here.
We begin with the following fundamental result which is established in [2]
and which examines when the Jacobi operator or the skew-symmetric curvature
operator commutes with the Ricci operator:
1 Lemma. The following conditions are equivalent for a model M:
(1) J(v)ρ = ρJ(v) for all v ∈ V .
(2) R(v1, v2)ρ = ρR(v1, v2) for all v1, v2 ∈ V .
(3) A(ρv1, v2, v3, v4) = A(v1, ρv2, v3, v4) = A(v1, v2, ρv3, v4)
= A(v1, v2, v3, ρv4) for all v1, v2, v3, v4 ∈ V .
One says that a model M is decomposable if there is an orthogonal direct
sum decomposition V = V1 ⊕ V2 inducing a splitting A = A1 ⊕ A2; M is
indecomposable if it is not decomposable. Let Spec(ρ) ⊂  be the spectrum of
the Ricci operator. One has [3]:
2 Lemma. If M is an indecomposable Jacobi–Ricci commuting model, then
Spec(ρ) = {a1} or Spec(ρ) = {a1 ± a2
√−1} where a2 > 0.
Although ρ is self-adjoint, ρ need not be diagonalizable in the higher signa-
ture setting and in fact the Jordan normal form of ρ can be quite complicated. To
simplify the discussion, we shall suppose ρ complex diagonalizable henceforth.
Motivated by Lemmas 1 and 2, we make the following:
3 Definition. IfM is a model which has any of the (3) equivalent properties
listed in Lemma 1, then M is said to be a Jacobi–Ricci commuting model. If in
addition, the Ricci operator ρ is complex diagonalizable and Spec(ρ) = {a1} or
Spec(ρ) = {a1± a2
√−1} for a2 > 0, then M is said to be a simple Jacobi–Ricci
commuting model. If M is a pseudo-Riemannian manifold, then M is said to be
simple Jacobi–Ricci commuting if M(M, P ) is a simple Jacob–Ricci commuting
model for all points P of M .
The following is immediate from the definitions we have given.
____________________________________________________________________________
Simple Jacobi–Ricci commuting algebraic curvature tensors 343
4 Lemma. Let M be a model.
(1) M is Einstein if and only if M is a simple Jacobi–Ricci commuting model
with Spec(ρ) = {a1}.
(2) Let M be a simple Jacobi–Ricci commuting model which is not Einstein.
Set J = JM := a−12 {ρ− a1 id}. Then J is a self-adjoint complex structure
on V , A(Jx, y, z, w) = A(x, Jy, z, w) = A(x, y, Jz,w) = A(x, y, z, Jw),
and ρ = a1 + a2J .
In view of Lemma 4 (1), we shall assume M is not Einstein henceforth. The
following ansatz for constructing simple Jacobi–Ricci commuting models which
are not Einstein will be crucial:
5 Definition. Let N := (V0, g, A1, A2) where g is a positive definite inner
product on a finite dimensional real vector space V0 and where A1 and A2 are
Einstein algebraic curvature tensors with Einstein constants, respectively, a1
and a2 > 0. Extend g, A1, and A2 to be complex linear on the complexification
V

:= V0⊗  . Let V := V0⊕V0
√−1 be the underlying real vector space of V

,
let 〈·, ·〉 := Re g(·, ·), and let A := Re{A1+
√−1A2} defineM(N) := (V, 〈·, ·〉, A).
The following classification result is the fundamental result of this paper:
6 Theorem. Adopt the notation established above:
(1) M(N) is a simple Jacobi–Ricci commuting model which is not Einstein,
which has Spec{ρ} = {2a1 ± 2a2
√−1}, and which has that JM(N) is mul-
tiplication by
√−1.
(2) Let M be a simple Jacobi–Ricci commuting model which is not Einstein
with Spec(ρ) = {2a1 ± 2a2
√−1} for a2 > 0. Then there exists N so M is
isomorphic to M(N).
(3) If N = (V0, g, A1, A2) and N˜ = (V˜0, g˜, A˜1, A˜2), then M(N) is isomorphic
to M(N˜) if and only if there is an isomorphism θ : V0 → V˜0 and a skew-
___________________________________________________________________________
344 P. Gilkey, S. Nikcˇevic´
adjoint linear transformation T of (V0, g) with |T | < 1 so that:
g˜(θv, θw) = g(v,w) − g(Tv, Tw),
A˜1(θv, θw, θx, θy) = A1(v,w, x, y) −A1(Tv, Tw, x, y)
−A1(Tv,w, Tx, y) −A1(Tv,w, x, Ty) −A1(v, Tw, Tx, y)
−A1(v, Tw, x, Ty) −A1(v,w, Tx, Ty) +A1(Tv, Tw, Tx, Ty)
−A2(Tv,w, x, y) −A2(v, Tw, x, y) −A2(v,w, Tx, y)
−A2(v,w, x, Ty) +A2(Tv, Tw, Tx, y) +A2(Tv, Tw, x, Ty)
+A2(Tv,w, Tx, Ty) +A2(v, Tw, Tx, Ty)
A˜2(θv, θw, θx, θy) = A2(v,w, x, y) −A2(Tv, Tw, x, y)
−A2(Tv,w, Tx, y) −A2(Tv,w, x, Ty) −A2(v, Tw, Tx, y)
−A2(v, Tw, x, Ty) −A2(v,w, Tx, Ty) +A2(Tv, Tw, Tx, Ty)
+A1(Tv,w, x, y) +A1(v, Tw, x, y) +A1(v,w, Tx, y)
+A1(v,w, x, Ty) −A1(Tv, Tw, Tx, y) −A1(Tv, Tw, x, Ty)
−A1(Tv,w, Tx, Ty) −A1(v, Tw, Tx, Ty)
(2)
Theorem 6 completes the analysis in the algebraic setting. In the geometric
setting, by contrast, the situation is still far from clear. However there is a
geometrical example known [2] in signature (2, 2) which may be described as
follows; we refer to [2] for further details. Let (x1, x2, x3, x4) be coordinates
on  4. Define a metric whose non-zero components are, up to the usual 2
symmetries, given by:
g(∂1, ∂3) = g(∂2, ∂4) = 1, g(∂3, ∂4) = s(x22 − x21),
g(∂3, ∂3) = 2sx1x2, g(∂4, ∂4) = −2sx1x2 . (3)
7 Lemma. Let M be as in Equation (3). Then M is a locally symmetric
simple Jacobi–Ricci commuting manifold with Spec(ρ) = {±2s√−1} of signa-
ture (2, 2). The Ricci operator and non-zero curvatures are described by:
R1314 = s, R1323 = −s, R1424 = s, R2324 = −s,
ρ∂1 = −2s∂2, ρ∂2 = 2s∂1, ρ∂3 = 2s∂4, ρ∂4 = −2s∂3 .
The remainder of this note is devoted to the proof of Theorem 6.
Proof of Theorem 6 (1). We generalize the discussion of [2]. Let {ei} be
an orthonormal basis for V0. Let e+i := ei and e
−
i :=
√−1ei be an orthonormal
basis for V ; the vectors e+i are spacelike and the vectors e
−
i are timelike so
M(N) has neutral signature. Clearly the symmetries of Equation (1) hold for
the complexification of A1 and A2 and, consequently, for A1 +
√−1A2 and
A = Re(A1 +
√−1A2). Thus M(N) is a model and the non-zero components of
____________________________________________________________________________
Simple Jacobi–Ricci commuting algebraic curvature tensors 345
A relative to this basis are given by:
A(e−i , e
+
j , e
+
k , e
+
l ) = A(e
+
i , e
−
j , e
+
k , e
+
l ) = A(e
+
i , e
+
j , e
−
k , e
+
l )
= A(e+i , e
+
j , e
+
k , e
−
l ) = −A2(ei, ej , ek, el),
A(e+i , e
−
j , e
−
k , e
−
l ) = A(e
−
i , e
+
j , e
−
k , e
−
l ) = A(e
−
i , e
−
j , e
+
k , e
−
l )
= A(e−i , e
−
j , e
−
k , e
+
l ) = A2(ei, ej , ek, el),
A(e+i , e
+
j , e
+
k , e
+
l ) = A(e
−
i , e
−
j , e
−
k , e
−
l ) = A1(ei, ej , ek, el),
A(e+i , e
+
j , e
−
k , e
−
l ) = A(e
+
i , e
−
j , e
+
k , e
−
l ) = A(e
−
i , e
+
j , e
+
k , e
−
l )
= A(e+i , e
−
j , e
−
k , e
+
l ) = A(e
−
i , e
+
j , e
−
k , e
+
l ) = A(e
−
i , e
−
j , e
+
k , e
+
l )
= −A1(ei, ej , ek, el) .
(4)
Let ρ := ρA and ρi := ρAi . We sum over k in the following expansions to see:
〈ρe+i , e+j 〉 = A(e+i , e+k , e+k , e+j )−A(e+i , e−k , e−k , e+j ) = 2ρ1(ei, ej) = 2a1δij ,
〈ρe−i , e−j 〉 = A(e−i , e+k , e+k , e−j )−A(e−i , e−k , e−k , e−j ) = −2ρ1(ei, ej) = −2a1δij,
〈ρe+i , e−j 〉 = A(e+i , e+k , e+k , e−j )−A(e+i , e−k , e−k , e−j ) = −2ρ2(ei, ej) = −2a2δij.
This shows that ρe±i = 2a1e
±
i ± 2a2e∓i . Thus we may view ρ as acting by
complex scalar multiplication by λ := 2a1 + 2a2
√−1 on V

. This implies that
the underlying real operator is complex diagonalizable and Spec(ρ) = {λ, λ¯}.
Since the tensors Ai were extended to be complex multi-linear, we have
A(ρv1, v2, v3, v4) = Re{A1(λv1, v2, v3, v4) +
√−1A2(λv1, v2, v3, v4)}
= Re{λA1(v1, v2, v3, v4) +
√−1λA2(v1, v2, v3, v4)}
= Re{A1(v1, λv2, v3, v4) +
√−1A2(v1, λv2, v3, v4)} = A(v1, ρv2, v3, v4) .
This establishes one equality of Lemma 1 (3); the other equalities follow similarly
and hence M(N) is Jacobi–Ricci commuting as well. QED
We shall need the following technical result before establishing the second
assertion of Theorem 6. Although well known, we include the proof for the sake
of completeness and to establish notation:
8 Lemma. Let J be a self-adjoint map of (V, 〈·, ·〉) so that J2 = − id. Then
there exists an orthonormal basis {e±1 , . . . , e±p } for V so that Je±i = ±e∓i .
Proof. We assume that p = 1 as the general result then follows by induc-
tion. Let {f±} be an orthonormal basis for V where f+ is spacelike and f− is
timelike. As J is trace-free and self-adjoint,
J =
(
a b
−b −a
)
.
___________________________________________________________________________
346 P. Gilkey, S. Nikcˇevic´
Since J2 = − id, b2 − a2 = 1. Let e(θ) := cosh θf+ + sinh θf−. Then
〈Je(θ), e(θ)〉
= 〈(a cosh θ + b sinh θ)f+ + (−b cosh θ − a sinh θ)f−, cosh θf+ + sinh θf−〉
= a cosh2 θ + b cosh θ sinh θ + b cosh θ sinh θ + a sinh2 θ
= 12{(a + b)e2θ + (a− b)e−2θ} .
Since a2 − b2 = −1, a+ b and a− b have opposite signs. Thus for some value of
θ, we have 〈Je(θ), e(θ)〉 = 0. Set e+i = e(θ) and e−i = Je(θ). QED
Proof of Theorem 6 (2). Let M = (V, 〈·, ·〉, A) be a simple Jacobi–Ricci
commuting model. Assume that the Ricci operator ρ is complex diagonalizable
and that Spec(ρ) = {2a1 ± 2a2
√−1} for a2 > 0. By Lemma 8, there is an
orthonormal basis {e±i } for V so Je±i = ±e∓i . Set
A1(ei, ej , ek, el) := A(e+i , e
+
j , e
+
k , e
+
l ),
A2(ei, ej , ek, el) := −A(e−i , e+j , e+k , e+l ) .
We may then derive the relations of Equations (4) from Lemma 4 (2). We check
that A1 and A2 are algebraic curvature tensors by verifying that:
A1(ei, ej , ek, el) = A(e+i , e
+
j , e
+
k , e
+
l ) = −A(e+j , e+i , e+k , e+l )
= −A1(ej , ei, ek, el),
A1(ei, ej , ek, el) = A(e+i , e
+
j , e
+
k , e
+
l ) = A(e
+
k , e
+
l , e
+
i , e
+
j )
= A1(ek, el, ei, ej),
A1(ei, ej , ek, el) +A1(ej , ek, ei, el) +A1(ek, ei, ej , el)
= A(e+i , e
+
j , e
+
k , e
+
l ) +A(e
+
j , e
+
k , e
+
i , e
+
l ) +A(e
+
k , e
+
i , e
+
j , e
+
l ) = 0,
A2(ei, ej , ek, el) = −A(e−i , e+j , e+k , e+l ) = A(e+j , e−i , e+k , e+l )
= −A(e−j , e+i , e+k , e+l ) = −A2(ej , ei, ek, el),
A2(ek, el, ei, ej) = −A(e−k , e+l , e+i , e+j ) = −A(e+i , e+j , e−k , e+l )
= −A(e−i , e+j , e+k , e+l ) = A2(ei, ej , ek, el),
A2(ei, ej , ek, el) +A2(ej , ek, ei, el) +A2(ek, ei, ej , el)
= −A(e−i , e+j , e+k , e+l )−A(e−j , e+k , e+i , e+l )−A(e−k , e+j , e+i , e+l )
= −A(e−i , e+j , e+k , e+l )−A(e+j , e+k , e−i , e+l )−A(e+k , e+j , e−i , e+l ) = 0.
We verify A1 and A2 are Einstein by summing over k to compute
(ρ1ei, el) = A(e+i , e
+
j , e
+
j , e
+
l )
= 12A(e
+
i , e
+
j , e
+
j , e
+
l )− 12A(e+i , e−j , e−j e+l ) = 12 〈ρe+i , e+l 〉 = a1δil.
(ρ2ei, el) = −A(e−i , e+j , e+j , e+l )
= −12A(e−i , e+j , e+j , e+l ) + 12A(e−i , e−j , e−j e+l ) = −12〈ρe−i , e+l 〉 = a2δil.
____________________________________________________________________________
Simple Jacobi–Ricci commuting algebraic curvature tensors 347
The desired result now follows. QED
Replacing e−i by −e−i in Equation (4) yields an isomorphism between the
models N(V, (·, ·), A1, A2) and N(V, (·, ·), A1 ,−A2); it is for this reason that we
may always assume the Einstein constant of A2 is positive. This reflects that
complex conjugation defines a field isomorphism of  taking λ → λ¯ or, equiv-
alently, by replacing a2 by −a2 in the construction. More important, however,
is the fact that the splitting V = V+ ⊕ V− where V± := Span{e±i } which is
crucial to our discussion is highly non-unique. Let N = (V0, g, A1, A2) and let
N˜ = (V˜0, g˜, A˜1, A˜2). Let M = M(N) and M˜ = M(N˜). Let J and J˜ be the asso-
ciated complex structures on V and on V˜ , respectively. We then have maximal
spacelike subspaces V+ := V0 and V˜+ := V˜0 of V and V˜ , respectively, so that
for all x, y, z, w in V0 and for all x˜, y˜, z˜, w˜ in V˜0,
V+ ⊥ JV+, V˜+ ⊥ J˜ V˜+,
A1(v,w, x, y) = A(v,w, x, y), A˜1(v˜, w˜, x˜, y˜) = A˜(v˜, w˜, x˜, y˜),
A2(v,w, x, y) = A(Jv,w, x, y), A˜2(v˜, w˜, x˜, y˜) = −A˜(J˜ v˜, w˜, x˜, y˜),
g(x, y) = 〈x, y〉, g˜(x˜, y˜) = 〈x˜, y˜〉 .
Suppose that Θ is an isomorphism from M to M˜. We may then identify
V = V˜ and J = J˜ . The decomposition V = V+ ⊕ JV+ defines orthogonal
projections π±. Since V˜+ is spacelike, π+ defines an isomorphism θ from V˜+ to
V+. Let
T = −J ◦ π− ◦ θ−1 : V+ → V˜+ → V− → V+ .
We may then represent any element of V˜+ in the form v + JTv for v ∈ V+.
9 Lemma. Adopt the notation established above:
(1) V˜+ ⊥ JV˜+ if and only if T is skew-adjoint.
(2) The induced metric on V˜+ is positive definite if and only if |T | < 1.
Proof. We have that J is self-adjoint and that J2 = − id. Consequently,
we have the following implications which establish Assertion (1):
V˜+ ⊥ JV˜+.
⇔ 〈v + JTv, Jw + JJTw〉 = 0 for all v,w ∈ V+.
⇔ −〈v, Tw〉 − 〈Tv,w〉 = 0 for all v,w ∈ V+.
⇔ T is skew-adjoint.
We argue similarly to prove Assertion (2):
___________________________________________________________________________
348 P. Gilkey, S. Nikcˇevic´
〈v + JTv, v + JTv〉 > 0 for all 0 	= v ∈ V+.
⇔ g(v, v) − g(Tv, Tv) > 0 for all 0 	= v ∈ V+.
⇔ |Tv|2 < |v|2 for all 0 	= v ∈ V+.
⇔ |T | < 1. QED
Proof of Theorem 6 (3). Suppose Θ : M(V0, A1, A2) → M(V˜0, A˜1, A˜2)
is an isomorphism. We use Θ to identify V with V˜ and to parametrize V˜+ in
the form {v + JTV } where T is a skew-adjoint linear map of V0 with |T | < 1.
We then the following identities for all v, w, x, and y:
g˜(v,w) = 〈v + JTv,w + JTw〉,
A˜1(v,w, x, y) = A(v + JTv,w + JTw, x + JTx, y + JTy),
A˜2(v,w, x, y) = A(J(v + JTv), w + JTw, x + JTx, y + JTy) .
(5)
Lemma 1 (2) and Equation (5) imply that Equation (2) holds. This establishes
one implication of Theorem 6 (3). As the arguments are reversible, the converse
implication holds as well. QED
Acknowledgements. Research of P. Gilkey partially supported by the
Max Planck Institute in the Mathematical Sciences (Leipzig) and by Project
MTM2006-01432 (Spain). Research of S. Nikcˇevic´ partially supported by Project
144032 (Srbija).
References
[1] M. Brozos-Va´zquez, B. Fiedler, E. Garc´ıa–Rı´o, P. Gilkey, S. Nikcˇevic´, G.
Stanilov, Y. Tsankov, R. Va´zquez-Lorenzo, V. Videv: Stanilov–Tsankov–Videv
Theory, SIGMA 3 (2007), 095, 13 pages.
[2] P. Gilkey, S. Nikcˇevic´: Pseudo-Riemannian Jacobi–Videv manifolds, International
Journal of Geometric Methods in Modern Physics, 4 (2007), 727–738.
[3] P. Gilkey, V. Videv: Jacobi-Jacobi commuting models and manifolds, J. Geom. 92
(2009), 60–68.
[4] G. Stanilov: Curvature operators based on the skew-symmetric curvature operator and
their place in the Differential Geometry, preprint.
[5] G. Stanilov: Higher order skew-symmetric and symmetric curvature operators, C. R.
Acad. Bulg. Sci. 57 (2004), 9–13.
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