\\n\\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\nThank you all for being part of the journey. 5,000 times thank you!
\\n\\nNow with 5,000 titles available Open Access, which one will you read next?
\\n\\nRead, share and download for free: https://www.intechopen.com/books
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\nDr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\n\nThank you all for being part of the journey. 5,000 times thank you!
\n\nNow with 5,000 titles available Open Access, which one will you read next?
\n\nRead, share and download for free: https://www.intechopen.com/books
\n\n\n\n
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The main concepts of the book are scope innovations in accelerated tooth movement, new developments such as corticotomy, microperforations (MOP), piezosicion, photobiostimulation, laser in orthodontics, chemical agents, as well as complications and risks. The book contains interdisciplinary managements involving surgery first, cleft lip and palate therapy, orthognathic surgery, and obstructive sleep apnea. This internationally-recognized specialty is continuing to experience advancements in technology, instrumentation, and treatment methods.",isbn:"978-1-78985-182-3",printIsbn:"978-1-78985-181-6",pdfIsbn:"978-1-83962-116-1",doi:"10.5772/intechopen.73937",price:119,priceEur:129,priceUsd:155,slug:"current-approaches-in-orthodontics",numberOfPages:206,isOpenForSubmission:!1,hash:"2c77384eeb748cf05a898d65b9dcb48a",bookSignature:"Belma Işık Aslan and Fatma Deniz Uzuner",publishedDate:"April 10th 2019",coverURL:"https://cdn.intechopen.com/books/images_new/7139.jpg",keywords:null,numberOfDownloads:9749,numberOfWosCitations:3,numberOfCrossrefCitations:4,numberOfDimensionsCitations:9,numberOfTotalCitations:16,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"April 9th 2018",dateEndSecondStepPublish:"April 30th 2018",dateEndThirdStepPublish:"June 29th 2018",dateEndFourthStepPublish:"September 17th 2018",dateEndFifthStepPublish:"November 16th 2018",remainingDaysToSecondStep:"3 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:"Edited by",kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"42847",title:"Dr.",name:"Belma",middleName:null,surname:"Işık Aslan",slug:"belma-isik-aslan",fullName:"Belma Işık Aslan",profilePictureURL:"https://mts.intechopen.com/storage/users/42847/images/7121_n.jpg",biography:"Dr. Belma Işik Aslan was born in 1976 in Ankara-TURKEY. After graduating from TED Ankara College in 1994 she attended to Gazi University, Faculty of Dentistry in Ankara. She completed her PhD in orthodontic education at Gazi University between 1999-2005. Dr. Işik Aslan stayed at the Providence Hospital Craniofacial Institude and Reconstructive Surgery in Michigan, USA for three months as an observer. She worked as a specialist doctor at Gazi University, Dentistry Faculty, Department of Orthodontics between 2005-2014. She was appointed as associate professor in January, 2014. Dr. Işik Aslan still works as an instructor at the same faculty. She has published a total of 26 articles, 4 book chapters, 34 conference proceedings both internationally and nationally. She is a member of the Turkish Orthodontic Society and Turkish Cleft Lip and Palate Society. She is married and has 2 children. Her knowledge of English is at an advanced level.",institutionString:"Gazi University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"4",totalChapterViews:"0",totalEditedBooks:"1",institution:null}],coeditorOne:{id:"172009",title:"Dr.",name:"Fatma Deniz",middleName:null,surname:"Uzuner",slug:"fatma-deniz-uzuner",fullName:"Fatma Deniz Uzuner",profilePictureURL:"https://mts.intechopen.com/storage/users/172009/images/7122_n.jpg",biography:"Dr. Deniz Uzuner was born in 1969 in Kocaeli-TURKEY. After graduating from TED Ankara College in 1986, she attended the Hacettepe University, Faculty of Dentistry in Ankara. \nIn 1993 she attended the Gazi University, Faculty of Dentistry, Department of Orthodontics for her PhD education. After finishing the PhD education, she worked as orthodontist in Ankara Dental Hospital under the Turkish Government, Ministry of Health and in a special Orthodontic Clinic till 2011. Between 2011 and 2016, Dr. Deniz Uzuner worked as a specialist in the Department of Orthodontics, Faculty of Dentistry, Gazi University in Ankara/Turkey. In 2016, she was appointed associate professor. Dr. Deniz Uzuner has authored 23 Journal Papers, 3 Book Chapters and has had 39 oral/poster presentations. She is a member of the Turkish Orthodontic Society. 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The primary function of the lungs is the exchange of gas occurring at the level of alveoli, which are arranged as acini in the lung parenchyma. There is strong need to understand the mechanisms of alveolar maintenance and repair because damage to this region underlies many chronic adult lung diseases, such as chronic obstructive pulmonary disease, idiopathic pulmonary fibrosis, acute respiratory failure in pneumonia, and acute respiratory distress syndrome. Additionally, insufficient development of alveoli results in various neonatal and childhood diseases including bronchopulmonary dysplasia [1]. Despite the pivotal role of the alveoli in the development of lung diseases, the pathogenesis of these various conditions is still largely unknown, and treatment options for patients remain limited.
There is clear evidence that environmental exposures and genetic predisposition contribute to the pathogenesis of idiopathic pulmonary fibrosis (IPF). IPF is defined as a specific form of progressive chronic fibrotic interstitial pneumonia, that is occurring mainly in older adults, and is limited to the lungs [2]. IPF remains relatively rare, with an estimated incidence of roughly 10 cases per 100,000 person-years. Nonetheless, IPF is a lethal lung disorder with a predicted survival of 3–6 years from the onset of symptoms. Most of the deaths among patients with IPF are due to respiratory failure or complicating comorbidities [3]. The pathogenesis of IPF is characterized by continuous insults or micro-lesions to the alveolar epithelium, which result in abnormal activation of both epithelial cells and fibroblasts. Finally, there is an alteration in the deposition of collagen, which contributes to the irreversible fibrosis typical of the disease [4]. Various risk factors have been identified in the development of IPF, that can be divided between intrinsic and extrinsic [5]. Intrinsic risk factors include genetics, aging, sex, lung microbiome [6, 7, 8, 9], while extrinsic risk factors comprise cigarette smoking, environmental exposures, and air pollution [10, 11]. Moreover, studies of familial clustering of pulmonary fibrosis provided evidence that IPF is associated with genetic susceptibility. Multiple genes can affect alveolar stability, for example, genes encoding surfactant proteins A and C, genes associated with enhanced cell senescence by disruption of telomerase function, with the integrity of the epithelial barrier, and with mutant desmosome proteins [12, 13, 14, 15].
Chronic obstructive pulmonary disease (COPD) is another chronic lung pathology, representing a serious and growing global health problem, as it is currently a leading cause of death worldwide [16]. COPD is a disease characterized by irreversible airflow reduction, associated with a decline in lung function and increased inflammatory response [17]. It represents a massive health problem, and it is estimated to affect around 200 million people worldwide, with a projected estimate towards further increase in the near future [18]. COPD is the result of the interaction between genetic susceptibility and environmental factors [19]. A well acknowledged genetic cause is α1-antitrypsin deficiency [20], while among environmental factors cigarette smoking represents the main cause; nonetheless, environmental pollution, occupational exposure to dust and fumes, and exposure to passive smoke can induce an increased risk in non-smokers, as well [21, 22]. Exposure to cigarette smoke, which contains a large number of pro-oxidant molecules [23], causes direct damage to the epithelial cells of the airways, leading to increased inflammation and activation of neutrophils, macrophages and lymphocytes in the airways [24]. There is currently no cure for COPD, but fortunately most symptoms can be treated and controlled mostly pharmacologically, at least delaying its progression and worsening. It represents indeed the most common indication for lung transplantation, that is the only conclusive therapeutic option for severe COPD, particularly in younger patients.
Cigarette smoking likely represents the single most significant risk factor for several lung conditions, and it is strongly associated with COPD and IPF, both familial and sporadic. Although observations about environmental risk factors have many biases and limitations [25], increasing knowledge on the underlying causes of lung diseases is evidencing how oxidative stress (OXS) and reactive oxygen species (ROS) play a crucial pathogenetic role (Figure 1).
Oxidative stress in lung pathogenesis. Intrinsic and extrinsic risk factors contribute to the progression of lung damage in the development of idiopathic pulmonary fibrosis (IPF) and obstructive pulmonary disease (COPD). In particular, cigarette smoking, environmental exposures, and air pollution induce an increase of reactive oxygen species (ROS) and oxidative stress condition, which play a crucial pathogenetic role in lung diseases. ROS, such as H2O2 and O2−, generated from NOX2 and NOX4, have a central role in the pathogenesis of pulmonary diseases. Indeed, ROS produced by these enzymes are involved in alveolar epithelial cell apoptosis, activation of inflammation, and induction of tissue fibrosis, that are all mechanisms underlying the progression of IPF and COPD. Figure was prepared using images from Servier Medical Art by Servier (
The lungs are indeed highly susceptible to ROS-induced injuries. ROS are commonly thought to be a harmful by-product generated in cellular systems. However, recent studies have suggested that ROS physiological levels regulate important biological functions in cellular processes [2, 26]. Normally, ROS are tightly controlled by enzymes and antioxidant molecules. Nonetheless, excessive ROS accumulation may occur under certain conditions, thus making detoxification by the antioxidant system difficult. The result is indeed a condition called OXS that can affect cell proliferation, differentiation, aging, and death [27]. Cigarette smoke is responsible for significant oxidant burden and decreased antioxidant capacity even in plasma [28, 29]. ROS produced from cigarette smoke, combustion of organic matter and gases, like ozone and nitrogen dioxide, are featured on the lung epithelium [30], and could decrease antioxidant defenses, increasing OXS in the lungs [31].
A disrupted function of the redox system can consequentially impact on key cell signaling pathways involved in disease progression. Conversely, several signals can alter the oxidative state of lung cells. For example, the lung is constantly exposed to biomechanical forces, such as fluid shear stress, cyclic stretch, and pressure, due to the blood flowing through the pulmonary vessels, and the distension of the lungs during the breathing cycle. It is indeed known that cells within the lung respond to these changes by activating signal transduction pathways that can also alter their redox state with pathophysiological consequences [32]. Particularly in the vasculature, the two types of biomechanical stimuli, such as frictional force known as shear stress (SS), or wall shear stress (WSS) that acts tangentially to the vessel, could determinate dysregulation of the cellular redox status, that in turn could have effects on intracellular signaling pathways involved in disease progression [33]. For example, exposure of endothelial cells to laminar SS can induce a suppression of ROS levels [34, 35]. Conversely, exposure of endothelial cells to WSS using an irregular flow induces an increase of ROS levels and a reduced bioavailability of the vasodilator molecule NO [36], which is involved in preventing the activation and adhesion of platelets and leukocytes to the wall of the injured vessel [37].
A significant role in the pathogenesis of COPD is precisely the imbalance of ROS production and antioxidant capacity [38]. Changes in the redox balance in the lungs and circulatory system, genetic polymorphisms, and activation of transcription factors, such as the nuclear factor kappa B (NF-κB), lead to the molecular pathogenesis of COPD [39, 40]. Oxidized proteins and lipid products, such as isoprostanes and carbonylated proteins, can be identified in exhaled air, bronchoalveolar lavage fluid, and lung tissue from patients with fibrotic lung diseases and COPD [41, 42]. Furthermore, clinical worsening of COPD is often associated with down-regulation of the antioxidant system, thus a possible therapeutic method for COPD could be the administration of redox-protective antioxidants [38]. Finally, it is possible that maintaining a balance between oxidant and antioxidant species in COPD affected smokers may slow down disease progression [43].
As discussed, smoking, occupational exposures like asbestos or silica, and radiation are the principal sources of OXS with overproduction of ROS, that could lead and contribute to pulmonary fibrosis [44]. Indeed, OXS is an important molecular mechanism underlying fibrosis in a variety of organs, including lungs. Bleomycin-induced pulmonary fibrosis, the most commonly used experimental animal model, has been shown to be associated with marked increase in the level of ROS, oxidized proteins, DNA and lipids [45]. Following lung injury three main mechanisms (i.e. inflammation, coagulation disturbances, and OXS) are involved and alter the lung interstitial cell compartment and extracellular matrix (ECM) homeostasis, resulting overall in pulmonary fibrosis. ROS can be produced by several cellular types involved in fibrosis including alveolar macrophages [46, 47, 48] and lung epithelial cells [49]. In particular, ROS generated from the mitochondria of stressed or damaged epithelial cells are very important; their mitochondrial dysfunction results in the generation and release of ROS, such as H2O2 and O2−, further enhancing OXS and cell damage [50]. As previously discussed, NAD(P)H oxidase is the main source of ROS, and isoforms NOX1, NOX2, and NOX4 have a central role in the pathogenesis of pulmonary fibrosis [51] (Figure 1). For example, NOX4 is strongly expressed in the hyperplastic alveolar epithelium of IPF patients [52], and ROS produced by NOX4 are involved in alveolar epithelial cell apoptosis. Continuous epithelial apoptosis further supports activation of inflammatory processes and cytokine release, including myofibroblast activating molecules, such as TGF-β1, PDGF, IL-1, and TNFα (Figure 1). There is also evidence of direct pathogenetic involvement of these enzymes in IPF, for example for NOX2: in fact, supporting data have shown that mice genetically deficient in NOX2 do not develop IPF after bleomycin or carbon nanotubes exposure [51, 53]. Finally, the interplay between oxidative stress and TGF-β1 signaling is of great importance in promoting fibrosis. In fact, TGF-β1 is the most profibrogenic protein and can directly stimulate NOX-mediated ROS production, while OXS in turn can activate latent TGF-β1, setting up a vicious profibrogenic positive feedback loop [54].
As mentioned previously, fibrosis and oxidative stress are linked to a dysregulation of cellular homeostasis and impaired alveolar structure in chronic lung diseases [55]. Despite the paramount importance of animal models in biomedical and clinical research, they often do not fully recapitulate the pathogenesis of human IPF [56]. Moreover, there is increasing social and political pressure on reducing animal experimentation, according to the 3R’s principle of replacement, reduction, and refinement. Furthermore, the associated costs of animal purchasing, housing, and handling cannot be ignored, as well [57]. Under this perspective, 3D cultures (such as organoids) and innovative microfluidic devices (such as “organs-on-chip”) represent useful platforms to perform significant investigations
In vitro models of lung pathology. Both organoids and lungs-on-chip allow the creation of 3D systems where complex cell–cell interactions and multi-cellular cultures are possible. Moreover, drug discovery and testing in these settings can provide important preliminary results in vitro. Several features, though, are better reproduced in specific 3D systems, for example the inclusion of an extracellular matrix, or the modelling of ontogenetic-like mechanisms better fit in organoid cultures. Conversely, biomechanical cues, gas and liquid interfaces, and immune cells response are more finely tunable with organs-on-chip technology. Figure was prepared using images from Servier Medical Art by Servier (
Three-dimensional culture systems offer multiple advantages for
The more complex example of organotypic 3D cultures is represented by organoids. Lung organoids are self-assembling structures of lung cell types that replicate cell–cell interaction, cell-ECM interaction, and organ structure and function at the microscale, as similar as possible to in vivo histological architecture. They can be used as models of both physiological and pathological settings. Strikoudis et al. have modelled pulmonary fibrosis in lung organoids to study Hermansky-Pudlak syndrome (HSP) [66]. IPF and HSP both are characterized by lung fibrosis, and are now considered as similar clinical entities, albeit with distinct etiology. Lung organoids were generated from embryonic stem cells (ESCs) with specific mutations that strongly predispose to HSP. The resulting organoids displayed a fibrotic phenotype, with an enhanced number of mesenchymal cells, and increased deposition of fibronectin and collagen. Interestingly, HSP organoids share a strong signature with lung samples from IPF patients, including the overexpression of interlukin-11 (IL-11), a key driver of the fibrotic process that is stimulated also from OXS [67]. This finding validates HSP lung organoids as a tool to study IPF and other lung diseases characterized by fibrosis [66]. Similarly, Wilkinson et al. have developed an organoid from induced pluripotent stem cell (iPSC)-derived fibroblasts functionalized with hydrogel beads, that acts as a 3D alveolar template within a rotating bioreactor [68]. Interestingly, they discovered that organoid formation was not possible in their conditions without the inclusion of fetal lung fibroblasts. Treatment of cultures with exogenous TGF-β1 consistently increased contraction, expression of Collagen 1 and α-SMA, and the emergence of fibroblastic foci within the treated organoid. This system showed features of tissue scarring similar to IPF, thus confirming the feasibility of organoid culture systems to model lung fibrosis. Moreover, these lung organoids can recapitulate even a more complex and representative lung microenvironment when cultured with endothelial and epithelial cells [68]. As an example, using lung organoids from patients with IPF, Surolia et al. described a 3D model to predict the invasive response of IPF fibroblasts to antifibrotic drugs therapy. They observed that inhibition of vimentin intermediate filaments assembly can reduce the invasiveness of lung fibroblasts derived from the majority of the IPF patients tested, uncovering a possible novel therapeutic target for pulmonary fibrosis [69].
Overall, these 3D self-assembled systems recapitulate numerous pathogenetic features of diseases, but nonetheless still show several limitations in their application as models, such as lack of vascular network, immune cells, and other supporting cells (Figure 2). These features need to be implemented to reach higher levels of physiological relevance for lung disease modelling [69].
In the last decade, the integration of advanced bioengineering approaches (e.g. 3D multicellular cultures) with microfluidic and microfabricated substrates has led to the development of devices called “organs-on-chip” [70]. These bioengineered tools allow fine control and tuning of the microenvironment architecture, media composition, and cell–cell interactions. The combination of lung cells and micro/nanoengineering devices gave rise to new
As discussed above, COPD is a syndrome defined by progressive and chronic airflow limitation, due to the fact that lungs become inflamed, damaged, and narrowed. The main cause is smoking, but others exist such as long-term exposure to harmful fumes or dust, and rare genetic conditions [43]. As for IPF, the animal models of COPD present some limitations. For example, modelling cigarette smoke exposure fails to recapitulate some major airway phenotypes of COPD, such as hyperplasia of basal and mucin-producing cells, and mucus plugging of the airways [76]. Before the advent of lung-on-chip technology, the best-established
In this regard, innovative approaches, such as microfluidic lungs-on-chip, have been developed in the last years and helped filling this gap. In 2016, Benam et al. developed the human lung “small airway-on-a-chip”, a microfluidic device that supports and drives full differentiation of a columnar, pseudostratified, mucociliary bronchiolar epithelium, composed of cells isolated from healthy individuals or people with COPD, underlined by a functional microvascular endothelium [81]. They demonstrated that COPD small airway chips recapitulate important features of the disease, such as selective cytokine hypersecretion and neutrophil recruitment from the vascular flow in response to epithelial activation by pathogen-like stimuli. Moreover, exposure of the healthy epithelium to interleukin-13 (IL-13) reconstituted the asthmatic phenotype that involves goblet cells hyperplasia, cytokine hypersecretion, and decreased ciliary function [82]. The same group improved this system by developing a “Breathing-Smoking Human Lung-on-Chip”, a novel device that consists of four components: a small airway on-chip, a smoke generating robot, a micro-respirator, and a control software that mimics human smoking and breathing. This smoking airway-on-a-chip system effectively recapitulated several key smoke-triggered molecular changes that are known to occur in lung epithelial cells, including increased OXS [83]. When human airway chips fabricated using cells from healthy donors were exposed to whole cigarette smoke, the authors observed a significant increase in the expression of the anti-oxidant gene heme oxygenase 1 (HMOX1), and increased phosphorylation of the transcription factor nuclear factor-like 2 (Nrf2). The latter induces expression of cytoprotective genes, including HMOX1, protecting cells from OXS and chemical toxicity. Furthermore, they identified new smoke-induced dysfunction, such as reduced ciliary beating, a novel biomarker of COPD disease, and studied the epithelial responses to smoke generated by electronic cigarettes [84]. However, the main limitation of this system is the absence of cellular stromal components.
As mentioned before, COPD represents a group of lung diseases that also include refractory severe asthma. In this regard, Nesmith et al. have designed a human airway musculature-on-a-chip with bronchiolar smooth muscle cells on an elastomeric thin film. To recapitulate asthmatic inflammation
Basic and translational research on lung biology and pathology can greatly benefit from the development of 3D
This work was supported by grant # 2015BN82FK from the Italian Ministry of Education, University and Research (MIUR) to Isotta Chimenti.
The authors declare no conflict of interest.
Immersions and submersions, which are special tools in differential geometry, also play a fundamental role in Riemannian geometry, especially when the involved manifolds carry an additional structure (such as contact, Hermitian and product structure). In particular, Riemannian submersions (which we always assume to have connected fibers) are fundamentally important in several areas of Riemannian geometry. For instance, it is a classical and important problem in Riemannian geometry to construct Riemannian manifolds with positive or non-negative sectional curvature. Riemannian submersions between Riemannian manifolds are important geometric structures. Riemannian submersions between Riemannian manifolds were studied by O’Neill [1] and Gray [2]. In [3], the Riemannian submersions were considered between almost Hermitian manifolds by Watson under the name of almost Hermitian submersions. In this case, the Riemannian submersion is also an almost complex mapping and consequently the vertical and horizontal distributions are invariant with respect to the almost complex structure of the total manifold of the submersion. The study of anti-invariant Riemannian submersions from almost Hermitian manifolds was initiated by Şahin [4]. In this case, the fibers are anti-invariant with respect to the almost complex structure of the total manifold. This notion extended to different total spaces see: [5, 6, 7, 8, 9, 10, 11, 12, 13, 14].
On the other hand, as a generalization of Riemannian submersion, horizontally conformal submersions are defined as follows [15]: Suppose that
for every
at
One can see that Riemannian submersions are very special maps comparing with conformal submersions. Although conformal maps do not preserve distance between points contrary to isometries, they preserve angles between vector fields. This property enables one to transfer certain properties of a manifold to another manifold by deforming such properties.
Recently, we introduced conformal anti-invariant submersions [20] and conformal semi-invariant submersions [21] from almost Hermitian manifolds, and gave examples and investigated the geometry of such submersions (see also [22, 23]). We showed that the geometry of such submersions is different from their counterpart anti-invariant Riemannian submersions and semi-invariant Riemannian submersions. In the present paper, we define and study conformal anti-invariant submersions from almost product Riemannian manifolds, give examples and investigate the geometry of the total space and the base space for the existence of such submersions.
Our work is structured as follows: Section 2 is focused on basic facts for conformal submersions and almost product Riemannian manifolds. The third section is concerned with definition of conformal anti-invariant submersions, investigating the integrability conditions of the horizontal distribution and the vertical distribution. In Section 4, we study the geometry of leaves of the horizontal distribution and the vertical distribution. In Section 5, we find necessary and sufficient conditions for a conformal anti-invariant submersion to be totally geodesicness. The last section, we give some examples of such submersions.
In this section we recall several notions and results which will be needed throughout the chapter.
Let
Then, we say that
Then we get
Thus
for any vector fields
Conformal submersions belong to a wide class of conformal maps that we are going to recall their definition, but we will not study such maps in this paper.
Note that we can write the last equation more sufficiently as
A point
Next, we recall the following definition from [18]. Let
The fundamental tensors of a submersion were introduced in [1]. They play a similar role to that of the second fundamental form of an immersion. More precisely, O’Neill’s tensors
where
for
for all
We see that the skew-symmetric part of
Let
defined by
for
for
Finally, we recall the following lemma from [15].
In this section, we define conformal anti-invariant submersions from an almost product Riemannian manifold onto a Riemannian manifold, investigating the geometry of distributions
Let
For
where
and
for
due to
for
Since
We now study the integrability of the distribution
for any
Further, from (15) we get
Using (9), (10) and if we take into account
Thus, from (12) and Lemma 2.2 we derive
Moreover, using (17), we obtain
which proves
From Theorem 3.1, we deduce the following characterization.
for
Now, taking
Hence
We say that a conformal anti-invariant submersion is a conformal Lagrangian submersion if
Since
which shows
Now, using (12) we obtain
Since
which tells that
In this section, we shall investigate the geometry of leaves of
Since
Moreover, using Definition 3.1 and (17) we obtain
which proves
From Theorem 4.1, we also deduce the following characterization.
Now, if we have (i) and (iii), then we obtain
Now, taking
Thus,
From above equation,
For conformal Lagrangian submersion, we have the following result.
Since
which shows
Now, using (12) we obtain
which tells that
For the totally geodesicness of the foliations of the distribution
Since
here we have used that
Moreover, using Definition 3.1 and (17), we obtain
which proves
From Theorem 4.3, we deduce the following result.
Now, if we have (i) and (iii), then we obtain
From above equation,
If
for
which shows
In this section, we shall examine the totally geodesicness of a conformal anti-invariant submersion. We give a necessary and sufficient condition for a conformal anti-invariant submersion to be totally geodesic map. Recall that a smooth map
It is obvious that if
Taking inner product in (31) with
From (32),
Again if
Taking inner product in (33) with
From above equation,
Then from (7) and (8) we arrive at
From above equation,
Since
Thus
Then, since
Here we present another result on conformal anti-invariant submersion to be totally geodesic.
for any
for any
for any
which gives our assertion.□
In this section, we now give some examples for conformal anti-invariant submersions from almost product Riemannian manifolds.
We say that a conformal anti-invariant submersion is proper if
and
where
and
where
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