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Historia Mathematica 36 (2009) 161–170 www.elsevier.com/locate/yhmat

GheorgheTi¸ ¸ teica and the origins of affine differential

Alfonso F. Agnew a, Alexandru Bobe b, Wladimir G. Boskoff b, Bogdan D. Suceava˘ a,∗

a Department of , P.O. Box 6850, California State University Fullerton, Fullerton, CA 92834-6850, USA b Department of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constan¸ta,

Abstract We describe the historical and ideological context that brought to the fore the study of a centro-affine invariant that subsequently received much attention. The invariant was introduced byTi¸ ¸ teica in 1907, and this discovery has been viewed by many as a consequence of Klein’s . We thus present the starting of affine , as it was discovered byTi¸ ¸ teica after his years in the Ph.D. program in Paris (1896–1899) under the guidance of Gaston Darboux. © 2008 Elsevier Inc. All rights reserved. Rezumat Descriem în prezenta nota˘ contextul istoricsi ¸ istoria ideilor care au condus la descoperirea invariantului centro-afin introdus de GheorgheTi¸ ¸ teica în 1907. Studiul acestui invariant poate fi considerat o consecin¸ta˘ a programului de la Erlangen. Prezentam˘ aici originea geometriei diferen¸tiale afine, a¸sa cum a fost conceputade¸˘ Ti¸teica dupa˘ anii lui de studii doctorale la Paris (1896–1899) sub conducerea lui Gaston Darboux. © 2008 Elsevier Inc. All rights reserved.

Keywords: Affine differential geometry; Centro-affine invariant; Tzitzeica surfaces; Affine ; Affine hyperspheres; Erlangen Program

GheorgheTi¸ ¸ teica introduced the first concepts of affine differential geometry,1 a research that is still attracting the interest of many researchers.2 There are several monographs that provide a survey of the area, including Li et al. [1993], Nomizu and Sasaki [1994], Su [1983]. In this note, we present pertinent details ofTi¸ ¸ teica’s life and his work on affine differential geometry. We emphasize his academic studies and his discovery of the centro-affine invariant that today bears his name—a discovery that spawned research in affine differential geometry. With this goal in mind, we focus our discussion mainly on the historical period 1892–1909.

* Corresponding author. E-mail addresses: [email protected] (A.F. Agnew), [email protected] (A. Bobe), [email protected] (W.G. Boskoff), [email protected] (B.D. Suceava).˘ 1 See, e.g., Ianu¸s [1995], Nomizu and Sasaki [1994, ix], Vranceanu˘ [1972]. 2 To mention a few more recent examples, see Baues and Cortés [2003], Calabi [1990], Cheng and Yau [1986], Dillen and Vrancken [1994], Gross and Siebert [2003], Loftin [2002], Loftin et al. [2005], Li et al. [1993], Oliker and Simon [1992], Opozda [2006], Scharlach et al. [1997], Scharlach and Vrancken [1996], Simon et al. [1991], Vrancken et al. [1991].

0315-0860/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.hm.2008.09.001 162 A.F. Agnew et al. / Historia Mathematica 36 (2009) 161–170

1. Biography

GheorgheTi¸ ¸ teica was born on October 4, 1873 in Turnu-Severin, Romania. His father died when he was 8 years old, and he supported himself in school by earning a scholarship, ranking first in his class every year.3 After graduating from high school, he was admitted at once to the ¸ScoalaNormala˘ Superioara˘ and as an undergraduate student with a major in mathematics in the Faculty of Sciences at the University of . He was able to pursue these studies after earning a scholarship. Among his professors were David Emmanuel, Spiru Haret, Constantin Gogu, and Iacob Lahovary.Ti¸ ¸ teica graduated fromScoala ¸ Normala˘ Superioara˘ in 1895. It was at this time that he obtained his first teaching position as a substitute teacher at Seminarul Nifon.4 One year later, after passing the credential examinations, he accepted an untenured position in a high school in Gala¸tiwhere he actually never taught. Supported by his salary while on leave, he went to Paris where he became a student at the École Normale Supérieure. In July 1897 he passed certifying exams in differential and integral calculus, mechanics, and astronomy. He ranked first in his cohort when he obtained a second undergraduate degree from the Sorbonne. These accomplishments entitled him to a full tuition waiver for his Ph.D. studies, as well as reimbursement for several other fees related to the undergraduate degree. Among the professors with whom he took classes were Henri Poincaré, Édouard Goursat, Gabriel Koenigs, Charles Hermite, Charles Emile Picard, Jules Tannery, Paul Appell, and last but not least, Gaston Darboux, who became Ti¸¸ teica’s Ph.D. thesis advisor.5 Ti¸¸ teica worked for two years on his Ph.D. thesis, Sur les congruences cycliques et sur les systèmes triplement conjugués, which he defended on June 30, 1899. His thesis committee consisted of Darboux, Koenigs, and Goursat. WhenTi¸ ¸ teica came to Paris in 1896, the fourth of Darboux’s treatise on the theory of surfaces had just been published [Darboux, 1972]. Darboux, who was by then a very influential mathematician (he was later appointed Dean of the Faculty of Sciences), agreed to work withTi¸ ¸ teica, who was then 23 years old. There is little doubt that Darboux’s approach to the geometry of surfaces had a major influence onTi¸ ¸ teica’s academic development:Ti¸ ¸ teica ended up incorporating Darboux’s viewpoint into his work to such an extent that he continued to refer to examples, formulas, and methods from his former advisor’s texts for the next three decades. To place this remark in context, we recall here what Eisenhart wrote about Darboux: “His writings possess not only content but singular finish and refinement of style. In the presentation of results, the form of exposition was carefully studied. Darboux’s varied powers combined with his personality in making him a great teacher, so that he always had about him a of able students” [Eisenhart, 1917/1918, 236]. It is of interest that in the fourth volume of Darboux [1972, 83–87], various classes of affine transformations are studied. It was in the spirit of the time to study differential objects from the Kleinian standpoint of transformation groups and, a few years later, as we shall describe below,Ti¸ ¸ teica found a relation between the geometry of surfaces and the centro-affine group, emphasizing one of the important of the Erlangen Program. Thus we shall see howTi¸ ¸ teica’s formative years played a role in anticipating his future discoveries. After defending his thesis,Ti¸ ¸ teica returned to Bucharest. After November 1, 1899, he substituted for Iacob Laho- vary, the professor of differential and integral calculus at the Faculty of Sciences, who was on leave. In May 1900, after the death of Gogu,Ti¸ ¸ teica became profesor agregat of and . In 1903 he became full professor of the same chair, a position he held until his death on February 5, 1939. As well as holding his position with the ,Ti¸ ¸ teica also held the chair of mathematical analysis at the Polytechnic School of Bucharest (substitute professor from October 1927, full professor from April 1928). During his career,Ti¸ ¸ teica wrote over one hundred papers, covering a large range of topics from elementary ge- ometry to advanced topics in differential geometry.6 A special chapter inTi¸ ¸ teica’s life is his contribution to Gazeta matematic˘a, a Romanian monthly publication that he and others founded in 1895, and that continues to be an im- portant influence in the Romanian educational system. His first work published there was [1895],followedby34 articles between 1895 and 1910. Together with three engineers, I. Ionescu, A. Ioachimescu, and V. Cristescu, he es- tablished the editorial board, and from the outset he took an active part in the editorial work. After his return from

3 For further biographical information onTi¸ ¸ teica, see Gheorghiu [1939], Ianu¸s [1995], Mihaileanu˘ [1955]. 4 Spiru Haret held a similar position at the beginning of his academic career. 5 For further information about Darboux, see, e.g., Alexander [1995]. 6 A comprehensive list of his works is presented in Pripoae and Gogu [2005]. A.F. Agnew et al. / Historia Mathematica 36 (2009) 161–170 163

Paris, he worked continuously with Gazeta matematic˘a, even during the periods when he was very active in research.7 As well as publishing notes, proposing problems, writing book reviews, and producing problem books published by the same editorial program as the journal,8 Ti¸¸ teica served continuously as its editor and referee until his death. In addition, he supervised its annual contest, a competition organized in a format similar to that of contemporary high school olympiads.9 He was always interested in publications that would reach large audiences, and he wrote papers on issues in mathematical education (for example [Ti¸¸ teica, 1903]). Additionally,Ti¸ ¸ teica published and edited, together with G. Longinescu, Natura, a general periodical covering a large array of scientific topics. He directed almost all the administrative and editorial work, while writing numerous notes intended for a wide readership.10 Possibly his most well-known book at the elementary level is the collection Probleme de geometrie [1962],of which six editions have appeared and which is still referred to today.11

2. The discovery of the centro-affine invariant

After he became full professor in Bucharest,Ti¸ ¸ teica had a very creative decade of research, publishing several important results in the period from 1906 to 1916. He was 33 years old when his first work on affine invariants [1907] appeared in the Comptes Rendues. In this article,Ti¸ ¸ teica starts his discussion on his centro-affine invariant by pointing out that the paper is about surfaces whose curvature is proportional to the fourth power of the distance from a fixed point O to the tangent drawn at M, where M is an arbitrary point of the surface. He calls any surface satisfying this curvature condition an S-surface, and he calls the fixed point O the center, which he uses as the coordinate origin. With this notation, he proves his first major result: If a linear transformation, which fixes both the plane at infinity and O,isappliedtoanS-surface, another S-surface with the same center O is obtained. In the last lines of the paper [1907, 1259], he states the main result: A linear transformation that changes neither the plane at infinity, nor the origin, leaves invariant the ratio between the Gaussian curvature of the surface and the fourth power of the distance from the origin to the tangent plane at a point of the surface. Ti¸¸ teica’s work [1907] was the paper that announced the result, even though many recent references, e.g., Loftin et al. [2005], Nomizu and Sasaki [1994], prefer to cite the two-part paper [Ti¸¸ teica, 1908c, 1909] as the original source. The first part of the paper [1908c], dated by the author October 30, 1907, and approved at the editorial meeting on November 10, 1907, includes the proofs of all the results stated in [1907]. It starts with the geometric motivation, the study of the deformations of the tetrahedral surfaces,12

Ax2/3 + By2/3 + Cz2/3 = 1, which leads to the study of S-surfaces, now often referred to in the literature asTi¸ ¸ teica surfaces. As a first step,Ti¸ ¸ teica uses the curvilinear coordinates introduced by Lelieuvre [1888] and described in the fourth volume of Darboux’s work [1972].¸Ti¸teica cites Darboux [1972, 24] as well as the classical reference [Bianchi, 1894, 162].Ti¸ ¸ teica then proves that, given a surface S with center O, its reciprocal polar with respect to a of center O is also an S-surface. Next he proves that any linear transformation transforms an S-surface into an S-surface. In [1909] Ti¸¸ teica replaces the definition of S-surfaces originally given in [1908c] by an affine definition. In the process of doing so, he obtains various characterizations of S-surfaces. He considers an asymptotic of an

7 Further publications byTi¸ ¸ teica not explicitly referred to in this article are [Ti¸¸ teica, 1906, 1908a, 1910a, 1910b, 1913a, 1913b, 1915a, 1915b, 1916, 1935]. 8 See our references forTi¸ ¸ teica, as well as Pripoae and Gogu [2005]; a more comprehensive list of references can be found from the Softwin database, cited in Ti¸¸ teica [1895], which contains the collection of Gazeta matematic˘a, 1895–2004. Furthermore,Ti¸ ¸ teica’s problem book in geometry [1962] was initially an educational project related to the editorial program developed with Gazeta matematic˘a. 9 It might be of interest to mention here that originally the Gazeta matematic˘a contest used to include oral examinations.Ti¸ ¸ teica wrote extensive columns in Gazeta in which he commented on contestants’ performances during the written and oral examinations. 10 ForTi¸ ¸ teica’s editorial and administrative work on journals, see Mihaileanu˘ [1955]. 11 A recent citation is in Pambuccian [2005]; see also Suceava˘ [2007]. 12 A tetrahedral surface is one whose equations are of the form x = D(u − a)m(v − a)n, y = E(u − b)m(v − b)n, z = F(u− c)m(v − c)n, where D,E,F,m,n are any constants [Eisenhart, 1909, 267]. The tetrahedral surfaces admit a conjugate network invariant through a continuous deformation [Ti¸¸ teica, 1956, 22, 25, 48, 50, 65]). They are called tetrahedral because for different values of m and n, the surfaces look like a deformed tetrahedron (see for example the astroidal sphere x2/3 + y2/3 + z2/3 = 1). 164 A.F. Agnew et al. / Historia Mathematica 36 (2009) 161–170

S-surface, given by u = constant. At every point of this line, he considers the tangent to another asymptotic line v = constant, which passes through the same point. All these tangents form a ruled surface R1, defined by the equations

∂x ∂y ∂z x = x + t ,y= y + t ,z= z + t , 1 ∂u 1 ∂u 1 ∂u where x1, y1, z1 are functions of v and t, along R1. He proves that the asymptotic development of any surface of type R1 is a cone having the origin as the . He then shows that this affine property characterizes the S-surfaces. He concludes by showing that a ruled surface is an S-surface if its asymptotic development reduces to a cone and if all of the osculating quadrics have as their center the vertex of the cone. It is of interest to point out thatTi¸ ¸ teica, in his contribution to the study of the geometry of groups of transformations, notably his proof that S-surfaces are invariants of a certain centro-affine group (see Vranceanu,˘ 197913), used Klein’s ideas concerning groups and geometries. Klein’s original viewpoint was that geometric transformation groups can be used to classify geometries andTi¸ ¸ teica was the first to apply Klein’s ideas from the Erlangen Program by using as group of transformations the centro-affine group and, as a consequence, to achieve important new results in affine differential geometry using Klein’s approach. Today this motivation is routinely included in standard texts on affine differential geometry, e.g., Su [1983, 1]. From the point of view of the history of ideas it is important to note that in the foreword to his book on the projective differential geometry of webs, which includes a discussion on his centro-affine invariant [Ti¸¸ teica, 1924, 254–256],Ti¸ ¸ teica mentions among the sources of his inspiration the works of Koenigs, Halphen and Wilczynski [Ti¸¸ teica, 1924, 3]:

From time to time, there are geometers, such as Mr. Koenigs, who prefer to study the infinitesimal elementary properties of a figure, which do not change under a projective transformation. Some have also considered certain analytical methods for the study of these properties. This is how Halphen introduced the seminal concept of projective differential invariants to study planar or space . However, Mr. Wilczynski deserves the credit for having created a projective theory, as complete as possible, of the infinitesimal properties of planar and space curves and of ruled surfaces. He has used for this purpose certain invariants and covariants of linear ordinary differential equations of third and fourth order and of systems of equations of second order. He has also applied a method similar to the study of surfaces and of congruences of lines in our space. [Ti¸¸ teica, 1924, 3]14

Ti¸¸ teica later gives [Ti¸¸ teica, 1924, 256] a precise reference to Wilczynski’s works [1907, 1908a, 1908b]. The foreword concludes with a methodical remark and a dedication to the memory of Gaston Darboux:

The calculations, which sometimes give the appearance of burdening the presentation, have behind them geometric intu- ition. Every equation has, almost always, an interpretation on the corresponding figure, such that one can easily follow the sequence of calculations. I learnt this method in the class and from the works of Darboux. Consequently, I would like to dedicate the present work to the memory of this incomparable master.15

13 Gh. Vranceanu˘ (1900–1979) was one of Ti¸teica’s successors in the field of differential geometry. His Ph.D. thesis, entitled Sopra un teorema di Weierstrass e le sue applicazioni alla stabilità, written under Tullio Levi-Civita’s supervision, was defended in 1924 in front of a committee of 11 examiners headed by Vito Volterra. 14 Ti¸¸ teica wrote: “De temps en temps, il y a eu des géomèters, comme M. Koenigs, qui ont étudié de préférence les propriétés des éléments infinitésimaux d’une figure, qui ne changent pas à la suite d’une transformation projective. On avait aussi considéré certaines méthodes analytiques pour la recherche de ces propriétés. C’est ainsi que Halphen avait introduit la notion féconde d’invariants différentiels projectifs pour l’étude des courbes planes ou gauches. Cependant, c’est à M. Wilczynski qui revient le mérite d’avoir créé une théorie projective, aussi complète que possible, des propriétés infinitésimales des courbes planes et gauches et des surfaces réglées. Il a employé à cet effet certains invariants et covariants des équations différentielles linéaires ordinaires du troisième et du quatrième ordre et des systèmes d’équations du second ordre. Il a appliqué aussi une méthode analogue à l’étude des surfaces et des congruences de droites de notre espace” [Ti¸¸ teica, 1924, 3]. 15 Ti¸¸ teica wrote: “ Les calculs, qui ont quelquefois l’air de charger l’exposition, ont à leur base l’intuition géométrique. Chaque équation a, presque toujours, une interprétation sur la figure correspondante, de sorte qu’on peut suivre aisément la suite des calculs. J’ai appris cette méthode au cours et dans les travaux de Darboux. Aussi je prends la permission de dédier cet ouvrage à la mémoire de ce maître incomparable” [Ti¸¸ teica, 1924, 4]. A.F. Agnew et al. / Historia Mathematica 36 (2009) 161–170 165

In the foreword of the Romanian of Ti¸¸ teica [1924], which was published in 1956, Gh. Vranceanu˘ wrote that even three decades after the first edition, the book is “one of the best in the area of projective differential theory of curves and surfaces,” which is why the Romanian Academy printed a new edition of the book. The concept ofTi¸ ¸ teica surfaces was first generalized to hypersurfaces in Euclidean spaces of arbitrary , under the slightly more general concept of an affine sphere.Thetermaffine hypersphere derives from the property that the lines passing through the affine normals at each point of an affine hypersphere intersect at one point—the center. When the point of intersection is at a finite distance, the affine sphere is called proper. This is the case corresponding to a¸Ti¸teica hypersurface. In Blaschke [1930, 212] it is pointed out that proper affine spheres were introduced and studied originally byTi¸ ¸ teica, and Blaschke indicates the sources of the theory in Ti¸¸ teica [1907, 1908b, 1908c, 1908d, 1909]. In a standard reference [Nomizu and Sasaki, 1994, 63], the connection between the notions of affine hypersphere andTi¸ ¸ teica invariant is presented as follows.16 For a so-called Blaschke hypersurface f : M → Rn+1, the Euclidean support function ρ¯ is defined for any point x ∈ M by ρ¯ =f(x),N, where N is the Euclidean unit normal with respect to the Euclidean inner product on the ambient space Rn+1. Denote by K¯ the Euclidean Gauss–Kronecker curvature ¯ 17 function, i.e., the product K = κ1 · κ2 · ··· ·κn of the principal curvatures at every point of the hypersurface. Then, Nomizu and Sasaki point out that the following theorem goes back to Ti¸¸ teica [1908c, 1909]: The immersion f is an affine hypersphere with the origin as its center if and only if K/¯ ρ¯n+2 is constant. For a survey on developments of affine differential geometry, the most comprehensive references are Nomizu and Sasaki [1994] and Simon [2000]. Among the classical references there is Calabi [1972]. Several other surveys are available; for example, Simon, referring to special topics not covered in his work [2000], mentions that in Liu et al. [1996], there are references to 14 surveys written by 12 authors in Dillen et al. [1996]. In its overview of affine differential geometry, the survey by Simon [2000] covers eight aspects: affine curves, basic structures on (including important contributions of Barthel, Nomizu, Norden, Opozda, and others), immersions (with important contributions of Nomizu, Pinkall, Dillen, and others), classical affine hypersurface theory, hypersurfaces with specific local properties, degenerate hypersurfaces, global affine differential geometry, and sub- manifolds of codimension greater than one. The Romanian historical connection with affine differential geometry is discussed in Andonie [1965] (see also [Cruceanu, 2005] and [Soare, 2005]), which, for completeness, we paraphrase in the following overview. Results in one-dimensional affine differential geometry can be traced back to A. Transon [1841] and his consid- eration of the affine normal to a . But it was not until about 70 years later that the study of affine properties of curves and surfaces began. For example, in 1906, Pick initiated the development of the theory of natural equations in the geometry of an arbitrary group of continuous transformations of the plane [Pick, 1906]. Using Lie , he gave a procedure for obtaining invariants. In particular, he generalized the notion of the of an arc of a curve, corresponding to the curvature of a curve in . Ti¸¸ teica was the first geometer to study affine spheres, using Euclidean invariants. From a mathematical perspective, it is interesting to note Cruceanu’s remarks that “in the fundamental equations obtained byTi¸ ¸ teica in 1907–1909, we find for the first time two conjugate linear connections which are not symbols of Christoffel for any covariant tensor of second order. Thus,Ti¸ ¸ teica anticipates the idea of affine connections which has been introduced much later by H. Weyl [1918]”[Cruceanu, 2005, 2]. Ti¸¸ teica continued to publish on related matters, and at the Fifth International Congress of Mathematicians (Cam- bridge), 1912, he raised the problem of the study of curves and surfaces with respect to different groups of transfor- mations in the Kleinian spirit [Ti¸¸ teica, 1912]. Indeed, a decade afterTi¸ ¸ teica’s discovery, many researchers, notably Berwald, Blaschke, Franck, Gross, Koning, Liebmann, Pick, Radon, Reidemeister, Salkowski, Thomsen, and Winter- nitz, worked in earnest on geometric problems with particular attention to invariance under the equiaffine group. Further developments on the study of affine surfaces appeared in the work of E. Cartan published in [1924],aswell as work published in the next decade by Slebodzi´ nski´ [1937, 1939]. The work of the Blaschke school was continued by other authors such as Kubota, Su Buchin, and Nakajima, who formed a Japanese school, and who published many papers in journals such as the Tohoku Mathematical Journal and the Japanese Journal of Mathematics.

16 For a similar reference, see also Simon [2000, 938]. 17 For the standard terminology, see also Chen [2000, 288]. 166 A.F. Agnew et al. / Historia Mathematica 36 (2009) 161–170

Guggenheimer [1963] has a chapter on affine differential geometry, which includes a classification list of equiaffine homogeneous surfaces, the latter being completed by Nomizu and Sasaki in [1991]. Calabi in his Bourbaki volume [1980] made several major contributions which dealt for the first time with global problems concerning noncompact affine hyperspheres. The last 20 years have seen great progress and development in affine differential geometry with many papers, monographs presenting important classical as well as modern results, new discoveries, and open problems being published.

3. Impact of the theory and further recognition

The invariant, as well as other ideas introduced byTi¸ ¸ teica, was studied and used for several decades by many influential mathematicans. For example, his works were cited by, among others, L.P. Eisenhart in [1921], G. Fubini in [1924, 1926, 1928], and L. Godeaux in [1932].¸Ti¸teica was invited to write chapters in influential books, e.g., Ti¸¸ teica [1926], a detail pointed out in Mihaileanu˘ [1955], and to write surveys on trends in contemporary research, e.g. [1931], which was based on the notes from his 1926 talk (one of several) at the Sorbonne, as mentioned in Ionescu-Bujor [1964]. Ti¸¸ teica participated in several international meetings. For example, in 1924, at the International Congress of Math- ematicians in Toronto,Ti¸ ¸ teica was one of the chairs of the Geometry section, together with F. Severi, É. Cartan, A. Demoulin, and F. Morley [Fields, 1924, 16]. In Romania, at the University Al. I. Cuza in Ia¸si, O. Mayer, starting with [1927], devoted a series of papers to the study ofTi¸ ¸ teica curves and surfaces, especially the case of ruled surfaces [1928, 1940a, 1940b].18 TheworkofMayer and Myller [1933] is a comprehensive study on curves in a centro-affine plane. In this paper, affine geometry appears as a separate branch of geometry and specific methods are introduced for its investigation. Along the same lines of research, Mayer published a memoir on the theory of surfaces in a hyperbolic centro-affine space [1934]. By means of the theory of differential forms used successfully by Gauss, Blaschke, and Fubini–Cech for the Euclidean, equiaffine, and projective case respectively, Mayer associated to a surface a quadratic form and a cubic form that in fact determine the surface up to a centro-affinity. He also studied remarkable classes of surfaces in which he found connections to Ti¸¸ teica surfaces. In the same year, I. Popa, in his Ph.D. thesis [1934, 1935a] constructed the hyperbolic centro-affine geometry of curves and in [1935b] the parabolic centro-affine geometry of curves and surfaces. Mayer studied also the centro-affine geometry of webs in a plane [1938]. Thus, the geometers A. Myller, O. Mayer, and I. Popa can be considered as the founders of a school of geometry at Ia¸si, which had as its basis the work ofTi¸ ¸ teica and which from the late 1920s made an important contribution to the progress of research in differential geometry. Ti¸¸ teica’s work and the developments of affine differential geometry due to the school from Ia¸sihave influenced the work of people from Romania and abroad. For example, Gh.Th. Gheorghiu pointed out new properties ofTi¸ ¸ teica curves and surfaces [see 1956, 1962] and discovered in [1959] a new class of surfaces which, in fact, is a generalization ofTi¸ ¸ teica surfaces. Also, in Gheorghiu and Popa [1959], he extended some results ofTi¸ ¸ teica to the n-dimensional case. A. Nicolescu studied in [1945] the parabolic centro-affine geometry of curves in a plane and in [1956, 1962] the centro- affine geometry of surfaces. Using as a starting pointTi¸ ¸ teica’s papers, D. Papuc and E. Munteanu sketched in [1960] a theory of surfaces in a parabolic centro-axial-affine space. Gh. Gheorghiev and I. Popa studied in [1960, 1961, 1962] the affine and centro-affine geometry of varieties of cones and discovered an important class of varieties of quadratic cones now called “Ti¸ ¸ teica varieties.” Furthermore, D. Papuc pursued in [1963a, 1963b] a study of submanifolds in a Klein space with a complete reducible linear group and, among other things, obtained results in the geometry of hyper- surfaces in a centro-affine space. Gh. Vranceanu˘ studied the properties ofTi¸ ¸ teica hypersurfaces in [1972, 1977, 1979], as did G.G. Vranceanu,˘ in [1997]. It would be worthwhile to mention here that G.Ti¸ ¸ teica was D. Barbilian’s doctoral thesis advisor.19 His thesis, entitled Canonical representation of the addition of hyperelliptic functions, was defended on January 29, 1929. Bar-

18 Some of the historical information mentioned in this paragraph appears also in Cruceanu [2005, 2–3]. 19 For a recent account of some of Barbilian’s work, see Boskoff and Suceava˘ [2007]. A.F. Agnew et al. / Historia Mathematica 36 (2009) 161–170 167 bilian’s most important contributions are the discovery of the so-called Barbilians metrization procedure in metric geometry and his work on geometries.20 After Spiru Haret’s death in 1913, the Romanian Academy electedTi¸ ¸ teica as a full member. He became general secretary of the Academy in 1929, and he held this position until his death ten years later [Gheorghiu, 1939].Ti¸ ¸ teica was also the Dean of Faculty of Sciences (July 1919 – February 1923) and he served many times as the President of the Mathematical Section of the Romanian Society of Sciences. In 1930 he was elected a corresponding member of the Maryland Academy of Sciences [Ionescu-Bujor, 1964], and in 1934 he became Doctor Honoris Causa of the University of Warsaw. Recently, the Balkan Journal of Geometry and Its Applications dedicated the first issue of its 10th volume (2005) to topics related toTi¸ ¸ teica’s research and legacy. DiscussingTi¸ ¸ teica’s long-lasting impact on Romanian mathematics, Ianu¸s [1995] refers toTi¸ ¸ teica as the “founder of the Romanian school of geometry.”21 Mihaileanu˘ comments onTi¸ ¸ teica’s towering reputation within the Romanian school of mathematics:

Ti¸¸ teica is the first Romanian mathematician that published a large number of scientific works, and his achievements have resulted in a long lasting legacy and the birth of a research area in differential geometry, all taking place while he pursued a dedicated life as educator, professor and researcher, whose influence is still evident today. [Mihaileanu,˘ 1955]

References

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20 Barbilian, under the pen name of Ion Barbu, is one of the most important poets in . 21 For other details onTi¸ ¸ teica’s life and work see, e.g., Andonie [1965], Dumitru [1981], Gheorghiu [1939], Ianu¸s [1995], Ionescu-Bujor [1964], Pripoae and Gogu [2005], Teleman [2005], Vranceanu˘ [1972, 1979]. 168 A.F. Agnew et al. / Historia Mathematica 36 (2009) 161–170

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