Curvature: From Nicole Oresme (1320-1382) to

Curvature: From Nicole Oresme (1320-1382) to Contemporary Interpretations Bogdan D. Suceav˘a California State University, Fullerton

Fall 2017 SoCal-Nev Section Meeting

Bogdan D. Suceav˘ a California State University, Fullerton

Curvature: From Nicole Oresme (1320-1382) to Contemporary In

The first time when we discuss the definition of the curvature of a curve is in our calculus classes. Let γ : I → R3 be a curve parametrized by arc length. The curvature of a curve is dT , κ = ds where T is the unit tangent vector field.



Summary: When was the concept of curvature introduced for the first time? Relation with Aristotle’s work.



Summary: When was the concept of curvature introduced for the first time? Relation with Aristotle’s work. Why was it difficult to extend the idea of curvature from curves to surfaces? Leonhard Euler’s idea.



Summary: When was the concept of curvature introduced for the first time? Relation with Aristotle’s work. Why was it difficult to extend the idea of curvature from curves to surfaces? Leonhard Euler’s idea. Curvature on hypersurfaces. Bang-Yen Chen’s Curvature Invariants and their relation to Nash’s Embedding Theorem. Zhiqin Lu’s Proof of Normal Scalar Curvature Conjecture.



Summary: When was the concept of curvature introduced for the first time? Relation with Aristotle’s work. Why was it difficult to extend the idea of curvature from curves to surfaces? Leonhard Euler’s idea. Curvature on hypersurfaces. Bang-Yen Chen’s Curvature Invariants and their relation to Nash’s Embedding Theorem. Zhiqin Lu’s Proof of Normal Scalar Curvature Conjecture. A natural duality between triangle geometry and inequalities on hypersurfaces.



A major scholastic event took place in the Western Europe from the middle of the 12th century into the 13th century.



A major scholastic event took place in the Western Europe from the middle of the 12th century into the 13th century. The Recovery of Aristotle = copying or re-translating of most of Aristotle’s books, from Greek or Arabic text into Latin.



A major scholastic event took place in the Western Europe from the middle of the 12th century into the 13th century. The Recovery of Aristotle = copying or re-translating of most of Aristotle’s books, from Greek or Arabic text into Latin. In this interval of time there have been translated 42 books from Corpus Aristotelicum.



A major scholastic event took place in the Western Europe from the middle of the 12th century into the 13th century. The Recovery of Aristotle = copying or re-translating of most of Aristotle’s books, from Greek or Arabic text into Latin. In this interval of time there have been translated 42 books from Corpus Aristotelicum. It is believed that only about a third from Aristotle’s work survived to us.



A major scholastic event took place in the Western Europe from the middle of the 12th century into the 13th century. The Recovery of Aristotle = copying or re-translating of most of Aristotle’s books, from Greek or Arabic text into Latin. In this interval of time there have been translated 42 books from Corpus Aristotelicum. It is believed that only about a third from Aristotle’s work survived to us. There are works whose existence is mentioned by other medieval authors, but never made it to the modern times.



Before the Recovery of Aristotle, there were only two of Aristotle’s books translated into Latin: Categories and On Interpretation.



Before the Recovery of Aristotle, there were only two of Aristotle’s books translated into Latin: Categories and On Interpretation. St. Augustine (354 – 430) describes in the Confessions (IV.xvi.28) how he was underwhelmed by a reading of the Categories at the age of 20.



Before the Recovery of Aristotle, there were only two of Aristotle’s books translated into Latin: Categories and On Interpretation. St. Augustine (354 – 430) describes in the Confessions (IV.xvi.28) how he was underwhelmed by a reading of the Categories at the age of 20. Boethius (c. 476 – 526) planned to translate into Latin and comment upon the whole Aristotelian corpus, and reconcile it with Plato as well, but only a fraction of the project (the logic) was completed.



Before the Recovery of Aristotle, there were only two of Aristotle’s books translated into Latin: Categories and On Interpretation. St. Augustine (354 – 430) describes in the Confessions (IV.xvi.28) how he was underwhelmed by a reading of the Categories at the age of 20. Boethius (c. 476 – 526) planned to translate into Latin and comment upon the whole Aristotelian corpus, and reconcile it with Plato as well, but only a fraction of the project (the logic) was completed. Hidden in Aristotle’s text - an important mathematical detail: the idea of continuous measure.



Aristotle - Categories, Chapter 6 (fragment)




”Of Quantity, one kind is discrete, and another continuous; the one consists of parts, holding position with respect to each other, but the other of parts, which have not that position.




”Of Quantity, one kind is discrete, and another continuous; the one consists of parts, holding position with respect to each other, but the other of parts, which have not that position. Discrete quantity is, as number and sentence, but continuous, as line, superficies, body, besides place and time. For, of the parts of number, there is no common term, by which its parts conjoin, as if five be a part of ten, five and five, conjoin at no common boundary, but are separated. Three, and seven, also conjoin at no common boundary, nor can you at all take a common limit of parts, in number, but they are always separated, whence number is of those things which are discrete.”



Aristotle - Categories, Chapter 6

(fragment) ”In like manner a sentence, for that a sentence is quantity is evident, since it is measured by a short and long syllable; but I mean a sentence produced by the voice, as its parts concur at no common limit, for there is no common limit, at which the syllables concur, but each is distinct by itself. A line, on the contrary, is continuous, for you may take a common term, at which its parts meet, namely, a point, and of a superficies, a line, for the parts of a superficies coalesce in a certain common term. So also you can take a common term in respect of body, namely, a line, or a superficies, by which the parts of body are joined.”



Aristotle - Categories, Chapter 6

(fragment) ”Of the same sort are time and place, for the present time is joined both to the past and to the future. Again, place is of the number of continuous things, for the parts of a body occupy a certain place, which parts join at a certain common boundary, wherefore also the parts of place, which each part of the body occupies, join at the same boundary as the parts of the body, so that place will also be continuous, since its parts join at one common boundary.”



The fact that Aristotle discussed continuous things attracted the attention of a scholar in the 14th century.



The fact that Aristotle discussed continuous things attracted the attention of a scholar in the 14th century. Nicole Oresme.



Notre-Dame de Paris was essentially complete by 1345. At the time when Oresme wrote his work, Notre Dame was new. Bogdan D. Suceav˘ a California State University, Fullerton


Who defined first the concept of curvature and how was it used for the first time?



Who defined first the concept of curvature and how was it used for the first time? First definition of curvature: bishop Nicole Oresme, Tractatus de configurationibus qualitatum et motuum (written between 1351 and 1355).



Who defined first the concept of curvature and how was it used for the first time? First definition of curvature: bishop Nicole Oresme, Tractatus de configurationibus qualitatum et motuum (written between 1351 and 1355). Dating the historical period in which Orseme wrote that treatise: Claudia Kren, in her doctoral dissertation The Questiones super de celo, of Nicole Oresme, in 1965, at Univ. Wisconsin.



Nicole Orseme’s definition: “Sit circulus maior cuius semidyameter sit AB et circulus minor cuius semidyameter sit AC. Si ergo semidyameter AB dupla est ad semidyametrum AC, curvitas minoris circuli erit duplo intensior curvitate maioris, et ita de aliis proportionibus et curvitatibus.” “Consider the circles with radii AB and AC . If the radius AB is twice the radius AC , then the curvature of the smaller circle is double the curvature of the larger circle, and so on for all the other proportions of the curvature.”



In a paper published in 1952, J. L. Coolidge (1873-1954) appreciates that the story of curvature is “unsatisfactory”, and he points out that “the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicolas Oresme, whose work was called to my attention by Carl Boyer,”



In a paper published in 1952, J. L. Coolidge (1873-1954) appreciates that the story of curvature is “unsatisfactory”, and he points out that “the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicolas Oresme, whose work was called to my attention by Carl Boyer,” then Coolidge comments: “Oresme conceived the curvature of a circle as inversely proportional to the radius; how did he find this out?”



In a paper published in 1952, J. L. Coolidge (1873-1954) appreciates that the story of curvature is “unsatisfactory”, and he points out that “the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicolas Oresme, whose work was called to my attention by Carl Boyer,” then Coolidge comments: “Oresme conceived the curvature of a circle as inversely proportional to the radius; how did he find this out?” We incline to the following hypothesis: by reading Aristotle and taking classes the the College de Navarre.



In a paper published in 1952, J. L. Coolidge (1873-1954) appreciates that the story of curvature is “unsatisfactory”, and he points out that “the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicolas Oresme, whose work was called to my attention by Carl Boyer,” then Coolidge comments: “Oresme conceived the curvature of a circle as inversely proportional to the radius; how did he find this out?” We incline to the following hypothesis: by reading Aristotle and taking classes the the College de Navarre. Oresme needed the concept of curvitas in his ”doctrine of configurations”.



Nicole Orseme was born around 1320 in the village of Allemagne, near Caen, today Fleury-sur-Orne.



Nicole Orseme was born around 1320 in the village of Allemagne, near Caen, today Fleury-sur-Orne.

Figure: Nicole Oresme, the only portrait we have.

The first certain fact in his biography is that he was a “bursar” of the College of Navarre from 1348 to October 4, 1356, when he became a Master. Bogdan D. Suceav˘ a California State University, Fullerton


The College of Navarre was established by Queen Joan I of Navarre in 1305 for the study of Arts, Philosophy and Theology for the students who could not afford to go to the University of Paris.



The College of Navarre was established by Queen Joan I of Navarre in 1305 for the study of Arts, Philosophy and Theology for the students who could not afford to go to the University of Paris. Oresme’s major was in Theology, but it is not known when he earned his degree of Master in Theology.



The College of Navarre was established by Queen Joan I of Navarre in 1305 for the study of Arts, Philosophy and Theology for the students who could not afford to go to the University of Paris. Oresme’s major was in Theology, but it is not known when he earned his degree of Master in Theology. As a student, he had to observe the code at the College of Navarre, where the students were required to speak and write only in Latin and all subjects had to be learned by rote, which was a learning technique quite common in that period.



The College of Navarre was established by Queen Joan I of Navarre in 1305 for the study of Arts, Philosophy and Theology for the students who could not afford to go to the University of Paris. Oresme’s major was in Theology, but it is not known when he earned his degree of Master in Theology. As a student, he had to observe the code at the College of Navarre, where the students were required to speak and write only in Latin and all subjects had to be learned by rote, which was a learning technique quite common in that period. Oresme studied, among others, with Jean Buridan and Albert of Saxony and as a scholar affiliated with this institution wrote his most important works, e.g. De proportionibus proportionum, which is of particular importance for the history of mathematics, or Ad pauca respicientes of interest for the history of ideas in celestial mechanics. Bogdan D. Suceav˘ a California State University, Fullerton


Orseme remained Master of the College until December 4, 1361, when he was forced to resign.



Orseme remained Master of the College until December 4, 1361, when he was forced to resign. On November 23, 1362 he became a canon of the Rouen Cathedral, and



Orseme remained Master of the College until December 4, 1361, when he was forced to resign. On November 23, 1362 he became a canon of the Rouen Cathedral, and on March 18, 1364 dean of the Cathedral.

Figure: The Rouen Cathedral. The interior today. Bogdan D. Suceav˘ a California State University, Fullerton


Cathedral in Rouen, built from 950 to 1478. Tallest building in the world from 1876 to 1880.



Figure: Charles V, King of France, 1364-1380.

Most of the work written by scholars in the 14th century was written for the most informed audience of their time. Bogdan D. Suceav˘ a California State University, Fullerton


Oresme was in that period King’s confessor and adviser, but his political influence is difficult to assess;



Oresme was in that period King’s confessor and adviser, but his political influence is difficult to assess; it is very likely that some of his works, including the pages written against the astrologers



Oresme was in that period King’s confessor and adviser, but his political influence is difficult to assess; it is very likely that some of his works, including the pages written against the astrologers are written with the hope of limiting the influences of astrology on the King and collaborators from his close circles.



Oresme was in that period King’s confessor and adviser, but his political influence is difficult to assess; it is very likely that some of his works, including the pages written against the astrologers are written with the hope of limiting the influences of astrology on the King and collaborators from his close circles. Some time before 1370 he became one of Charles V’s (1364-1380) chaplains and at the king’s request he translated from Latin into French



Oresme was in that period King’s confessor and adviser, but his political influence is difficult to assess; it is very likely that some of his works, including the pages written against the astrologers are written with the hope of limiting the influences of astrology on the King and collaborators from his close circles. Some time before 1370 he became one of Charles V’s (1364-1380) chaplains and at the king’s request he translated from Latin into French Aristotle’s Ethics Politics Economics.



De configurationibus has 93 chapters, and is composed of three parts.



De configurationibus has 93 chapters, and is composed of three parts. → In the first part (the first 40 chapters) Oresme sets up the ground for a doctrine of configurations, then he applies the doctrine to qualities, focusing on “entities” which are permanent or enduring in time. While discussing these elements, he suggests that his theory could explain numerous physical and psychological phenomena.



De configurationibus has 93 chapters, and is composed of three parts. → In the first part (the first 40 chapters) Oresme sets up the ground for a doctrine of configurations, then he applies the doctrine to qualities, focusing on “entities” which are permanent or enduring in time. While discussing these elements, he suggests that his theory could explain numerous physical and psychological phenomena. → In the second part (the next 40 chapters), Oresme describes how graphical representation can be applied to “entities that are successive”, in particular to motion. He concludes this part with several examples could be extended to psychological effects, in particular to perceptions that are described as magic.



De configurationibus has 93 chapters, and is composed of three parts. → In the first part (the first 40 chapters) Oresme sets up the ground for a doctrine of configurations, then he applies the doctrine to qualities, focusing on “entities” which are permanent or enduring in time. While discussing these elements, he suggests that his theory could explain numerous physical and psychological phenomena. → In the second part (the next 40 chapters), Oresme describes how graphical representation can be applied to “entities that are successive”, in particular to motion. He concludes this part with several examples could be extended to psychological effects, in particular to perceptions that are described as magic. → Finally, in the third part (the last 13 chapters), Oresme describes external geometrical figures used to represent qualities and motions. He also describes that by comparing the areas of such figures one may have a basis for the comparisons of different qualities and motions. Bogdan D. Suceav˘ a California State University, Fullerton


To fully describe his theory, Orseme begins his De configurationibus by the following clarification: “Every measurable thing except numbers is imagined in the manner of continuous quantity.”



To fully describe his theory, Orseme begins his De configurationibus by the following clarification: “Every measurable thing except numbers is imagined in the manner of continuous quantity.” Then he argues that qualities can be “figured”.



To fully describe his theory, Orseme begins his De configurationibus by the following clarification: “Every measurable thing except numbers is imagined in the manner of continuous quantity.” Then he argues that qualities can be “figured”. Oresme indicates Aristotle as his source of inspiration.



To fully describe his theory, Orseme begins his De configurationibus by the following clarification: “Every measurable thing except numbers is imagined in the manner of continuous quantity.” Then he argues that qualities can be “figured”. Oresme indicates Aristotle as his source of inspiration. One distinction appears in chapter I.xi, where Oresme talks about uniform vs. difform qualities, and this discussion is continued in I.xiv with a discussion of “simple difform difformity”, which is of two kinds: simple and composite. He then pursues in I.xv by describing four kinds of simple difform difformity, which are described by drawing graphs; it is a very interesting early discussion of convexity and concavity. After this extensive discussion, performed without any algebraic notations, Oresme approaches “surface quality”. In chapers I.xix, I.xx, and I.xxi, he introduces the curvature.



A particular case in this theory, one of these qualities represents curvature (chapter I.xx), endowed with ”both extension and intensity”.



A particular case in this theory, one of these qualities represents curvature (chapter I.xx), endowed with ”both extension and intensity”. → Oresme writes (in M. Clagett’s translation): “we do not know with what, or with regard to what, the intensity of curvature is measured. For now it appears to me that there are only two [possible] ways [to speak of the measure of curvature]. The first is that the increase in curvature is a function of its departure from straightness, i.e. of its distance from straightness. This is [to be measured] by the quantity of the angle constituted of a straight line and a curve, e.g. an angle of contingence or perhaps another angle also constructed from a straight line and a curve.”




The modern concept is: turning angle.




The modern concept is: turning angle. In modern geometry, the rate of change of the turning angle with respect to arc length is curvature. Bogdan D. Suceav˘ a California State University, Fullerton



The modern concept is: turning angle. In modern geometry, the rate of change of the turning angle with respect to arc length is curvature. κ = dT ds . Bogdan D. Suceav˘ a California State University, Fullerton


Figure: The discussion on convexity is related to the definition of curvature. From the 1968 edition (with Marshall Clagett’s translation). Bogdan D. Suceav˘ a California State University, Fullerton


Figure: Chapter II.xl On the difformity of joys. ”One ought to speak in the same way concerning a joy or a pleasure, which I suppose to be a certain quality extended in time and intended in degree.” Bogdan D. Suceav˘ a California State University, Fullerton


Figure: Convex and concave as kept in a manuscript from the late 15th or earlier 16th century. An image selected for the 1968 Univ of Wisconsin edition, with Marshall Clagett’s translation. Bogdan D. Suceav˘ a California State University, Fullerton


In classical differential geometry, the co-called fundamental theorem of curves states that



In classical differential geometry, the co-called fundamental theorem of curves states that if two single-valued continuous functions κ(s) and τ (s), for s > 0, are given, then there exists one and only one space curve, determined but for its position in space, for which s is the arc length, measured from an appropriate point on the curve, κ is the curvature, and τ is the torsion.



In classical differential geometry, the co-called fundamental theorem of curves states that if two single-valued continuous functions κ(s) and τ (s), for s > 0, are given, then there exists one and only one space curve, determined but for its position in space, for which s is the arc length, measured from an appropriate point on the curve, κ is the curvature, and τ is the torsion. Oresme writes: “every difform curvature is difform in a way different from that in which any other quality of another kind could be, and [so it is difform] with a strange, marvelous, diverse kind of difformity.” (End of chapter I.xx.)



To summarize: a 14th century scholar




gave a correct definition for curvature of planar curves,





tried to apply curvature to understand the behavior of real-life phenomena, and





tried to apply curvature to understand the behavior of real-life phenomena, and

produced in his research a statement that anticipates the fundamental theorem of curves in the plane.



Figure: The Black Death was one of the major historical events witnessed by Nicole Oresme at the beginning of the period when he started to write De configurationibus. In the image, plague victims being blessed, shown with symptoms from a late 14th-century manuscript Omne Bonum by James le Palmer. These direct experiences must have inspired Oresme to reflect on the intensity of perceptions, and to attempt to model such quantities by proportions. Metaphors and examples based on senses are everywhere in his mathematical work.



The historical period 1351-1355:



The historical period 1351-1355: → in those years, Geoffrey Chaucer, later considered the father of English literature, was still a child in London



The historical period 1351-1355: → in those years, Geoffrey Chaucer, later considered the father of English literature, was still a child in London → Giovanni Boccaccio wrote the Decameron, largely completed by 1352



The historical period 1351-1355: → in those years, Geoffrey Chaucer, later considered the father of English literature, was still a child in London → Giovanni Boccaccio wrote the Decameron, largely completed by 1352 → in Florence, Francesco Petrarch was working and was the first to coin the description of the “Dark Ages”



The historical period 1351-1355: → in those years, Geoffrey Chaucer, later considered the father of English literature, was still a child in London → Giovanni Boccaccio wrote the Decameron, largely completed by 1352 → in Florence, Francesco Petrarch was working and was the first to coin the description of the “Dark Ages” → one of the major historical chronicles describing that historical period are Jean Froissart’s Chronicles, describing in detail the battles from the One Hundred Year War, and the Black Death, impacting most of the Europe in the interval 1346-1353



If Oresme clearly reached the first recorded definition of curvature for planar curves and described the first possible applications, then why his work was not discussed and cited?



If Oresme clearly reached the first recorded definition of curvature for planar curves and described the first possible applications, then why his work was not discussed and cited? → one generation after him his influence faded, his work was not continued, his heritage was less understood



If Oresme clearly reached the first recorded definition of curvature for planar curves and described the first possible applications, then why his work was not discussed and cited? → one generation after him his influence faded, his work was not continued, his heritage was less understood → later authors, as Christiaan Huygens (1629-1695) and Isaac Newton (1642-1727) discovered and developed fundamental concepts by themselves, and did not build on Oresme’s heritage



If Oresme clearly reached the first recorded definition of curvature for planar curves and described the first possible applications, then why his work was not discussed and cited? → one generation after him his influence faded, his work was not continued, his heritage was less understood → later authors, as Christiaan Huygens (1629-1695) and Isaac Newton (1642-1727) discovered and developed fundamental concepts by themselves, and did not build on Oresme’s heritage → when mathematics benefitted from the important revolution in sciences after 1600, Oresme’s mathematical texts were forgotten,



If Oresme clearly reached the first recorded definition of curvature for planar curves and described the first possible applications, then why his work was not discussed and cited? → one generation after him his influence faded, his work was not continued, his heritage was less understood → later authors, as Christiaan Huygens (1629-1695) and Isaac Newton (1642-1727) discovered and developed fundamental concepts by themselves, and did not build on Oresme’s heritage → when mathematics benefitted from the important revolution in sciences after 1600, Oresme’s mathematical texts were forgotten, and e.g. by 1952 only few specialists knew of their existence



If Oresme clearly reached the first recorded definition of curvature for planar curves and described the first possible applications, then why his work was not discussed and cited? → one generation after him his influence faded, his work was not continued, his heritage was less understood → later authors, as Christiaan Huygens (1629-1695) and Isaac Newton (1642-1727) discovered and developed fundamental concepts by themselves, and did not build on Oresme’s heritage → when mathematics benefitted from the important revolution in sciences after 1600, Oresme’s mathematical texts were forgotten, and e.g. by 1952 only few specialists knew of their existence This story is told in a paper written with Isabel M. Serrano, see Notices of the American Mathematical Society, October 2015.



If Oresme clearly reached the first recorded definition of curvature for planar curves and described the first possible applications, then why his work was not discussed and cited? → one generation after him his influence faded, his work was not continued, his heritage was less understood → later authors, as Christiaan Huygens (1629-1695) and Isaac Newton (1642-1727) discovered and developed fundamental concepts by themselves, and did not build on Oresme’s heritage → when mathematics benefitted from the important revolution in sciences after 1600, Oresme’s mathematical texts were forgotten, and e.g. by 1952 only few specialists knew of their existence This story is told in a paper written with Isabel M. Serrano, see Notices of the American Mathematical Society, October 2015. On Nicole Oresme and applied mathematics: a work coming soon, with Isabel M. Serrano and Anael Verdugo.



1737: The year when it is published Isaac Newton’s volume A Treatise of the method of fluxions and infinite series, with its applications to the geometry of curve lines, translated from the latin original not yet published. Printed in London for T. Woodman at Camden’s Head in New Round Court in the Strand; and J. Millan next to Will’s Coffee House at the entrance into Scotland Yard. Original work written probably in 1671.





Leonhard Euler writes Recherches sur la courbure des surfaces presented to the Berlin Academy on September 8, 1763. Published in Memoires de l’academie des sciences de Berlin 16, 1767, pp. 119–143.





One of the figures responsible for L. Euler’s confusion from 1763: the catenoid.



Gauss’s Disquisitiones, from 1827, represented a major change in paradigm.



Gauss’s Disquisitiones, from 1827, represented a major change in paradigm. We owe to Gauss the definition of curvature for surfaces.



In the theory of surfaces in the three dimensional real space, the two important curvature invariants are the Gaussian curvature and the mean curvature. Suppose N is the unit normal at the point p on the surface M. Then we consider all the normal sections through N(p), i.e., all the planes that are supported by the line of the vector N(p). The normal sections generate a class of planar curves on the surface M. If we denote κ1 (p) to be the infimum taken of all the curvatures of the normal sections at p ∈ M, and denote κ2 (p) to be the supremum taken of all the curvatures of the normal sections at p ∈ M,



then the Gaussian curvature at p is K (p) = κ1 (p) · κ2 (p), and the mean curvature at p is 1 H(p) = [κ1 (p) + κ2 (p)]. 2 Bogdan D. Suceav˘ a California State University, Fullerton


In 1831, Sophie Germain published a very important work where she introduced the mean curvature and argued that this invariant captures important features of the shape. (Mémoire sur la courbure des surfaces, Journal f¨ ur die reine und andewandte Mathematik, Herausgegeben von A. L. Crelle, Siebenter Band, pp. 1–29, Berlin, 1831.)



In 1831, Sophie Germain published a very important work where she introduced the mean curvature and argued that this invariant captures important features of the shape. (Mémoire sur la courbure des surfaces, Journal f¨ ur die reine und andewandte Mathematik, Herausgegeben von A. L. Crelle, Siebenter Band, pp. 1–29, Berlin, 1831.) Sophie Germain was interested how the curvatures of normal sections are distributed around a point on a surface. In original: “Quelle que soit la diversité des surfaces, les courbures linéaires produites par l’intersection du plan normal qui prendroit successivement toutes les positions possibles autour d’un point donné, offrira toujours le même arrangement symétrique. Cet arrangement constitue ce que nous nommerons dorénavant: loi de distribution de la courbure autour de chaqun des points de la surface.”



Sophie Germain’s argument to look at the mean curvature: the circular cylinder has K = 0, but we perceive it as curved.



Here is a surface with constant K = 1, and with variable mean curvature (color coded in the figure):

Figure made with 3D-XplorMath-J. Bogdan D. Suceav˘ a California State University, Fullerton


Surfaces satisfying H = 0 at every point are called minimal surfaces.



Surfaces satisfying H = 0 at every point are called minimal surfaces. Here is a minimal surface (Chen-Gackstatter surface, discovered in 1982) with variable Gaussian curvature (color coded in the figure):

Figure made with 3D-XplorMath-J. Bogdan D. Suceav˘ a California State University, Fullerton


Given a manifold M, can we embed it into a given ambient space?



Given a manifold M, can we embed it into a given ambient space? ´ Cartan and Maurice Janet. Origin of the problem: Elie



Given a manifold M, can we embed it into a given ambient space? ´ Cartan and Maurice Janet. Origin of the problem: Elie Theorem Nash Embedding Theorem: Every Riemannian manifold can be isometrically embedded into some Euclidean space.



Given a manifold M, can we embed it into a given ambient space? ´ Cartan and Maurice Janet. Origin of the problem: Elie Theorem Nash Embedding Theorem: Every Riemannian manifold can be isometrically embedded into some Euclidean space.



What is the best possible embedding of a manifold into the Euclidean space?



What is the best possible embedding of a manifold into the Euclidean space? Explore the case of a surface:



What is the best possible embedding of a manifold into the Euclidean space? Explore the case of a surface: Consider a surface S in R3 . Principal curvatures at P ∈ S : k1 , k2 . Gaussian curvature: K (p) = k1 · k2 . 2 Mean curvature: H(p) = k1 +k 2 .



What is the best possible embedding of a manifold into the Euclidean space? Explore the case of a surface: Consider a surface S in R3 . Principal curvatures at P ∈ S : k1 , k2 . Gaussian curvature: K (p) = k1 · k2 . 2 Mean curvature: H(p) = k1 +k 2 . Remark: (k1 − k2 )2 ≥ 0, or, rewrite:



What is the best possible embedding of a manifold into the Euclidean space? Explore the case of a surface: Consider a surface S in R3 . Principal curvatures at P ∈ S : k1 , k2 . Gaussian curvature: K (p) = k1 · k2 . 2 Mean curvature: H(p) = k1 +k 2 . Remark: (k1 − k2 )2 ≥ 0, or, rewrite: k12 − 2k1 k2 + k22 ≥ 0, k12 + 2k1 k2 + k22 ≥ 4k1 k2 . In terms of curvature invariants, at any point P ∈ S: (k1 + k2 )2 ≥ 4K .



What is the best possible embedding of a manifold into the Euclidean space? Explore the case of a surface: Consider a surface S in R3 . Principal curvatures at P ∈ S : k1 , k2 . Gaussian curvature: K (p) = k1 · k2 . 2 Mean curvature: H(p) = k1 +k 2 . Remark: (k1 − k2 )2 ≥ 0, or, rewrite: k12 − 2k1 k2 + k22 ≥ 0, k12 + 2k1 k2 + k22 ≥ 4k1 k2 . In terms of curvature invariants, at any point P ∈ S: (k1 + k2 )2 ≥ 4K . In conclusion: H 2 (P) ≥ K (P). (We read: extrinsic ≥ intrinsic) Bogdan D. Suceav˘ a California State University, Fullerton


We established: H 2 (P) ≥ K (P). ( extrinsic ≥ intrinsic) Why is this important? A surface S ⊂ R3 is minimal if at ∀P ∈ S : H(P) = 0.

(The figure shows Scherk’s Saddle Tower.) The study of minimal surfaces is a central topic in differential geometry.



Since H 2 (P) ≥ K (P), we have obtained the following obstruction to minimality: If there exists a point P ∈ S such that K (P) > 0, then H(P) 6= 0.




A classical theorem in the geometry of surfaces:




A classical theorem in the geometry of surfaces: If S is compact, then there exists P ∈ S such that K (P) > 0. Based on the above obstruction, we have just proved that a compact surface in R3 can not be minimal.



We turn our attention now to some classical ideas from the differential geometry of smooth hypersurfaces.



We turn our attention now to some classical ideas from the differential geometry of smooth hypersurfaces. Let σ : U ⊂ R3 → R4 be a hypersurface given by the smooth map σ.



We turn our attention now to some classical ideas from the differential geometry of smooth hypersurfaces. Let σ : U ⊂ R3 → R4 be a hypersurface given by the smooth map σ. Let p be a point on the hypersurface.



We turn our attention now to some classical ideas from the differential geometry of smooth hypersurfaces. Let σ : U ⊂ R3 → R4 be a hypersurface given by the smooth map σ. Let p be a point on the hypersurface. ∂σ Denote σk (p) = ∂x , for all k from 1 to 3. k



We turn our attention now to some classical ideas from the differential geometry of smooth hypersurfaces. Let σ : U ⊂ R3 → R4 be a hypersurface given by the smooth map σ. Let p be a point on the hypersurface. ∂σ Denote σk (p) = ∂x , for all k from 1 to 3. k Consider {σ1 (p), σ2 (p), σ3 (p), N(p)}, the orthonormal Gauss frame of the hypersurface, where N denotes the normal vector field to the hypersurface at every point.



The quantities similar to κ1 and κ2 in the geometry of surfaces are the principal curvatures, denoted λ1 , λ2 , λ3 . They are introduced as the eigenvalues of the so-called Weingarten linear map.



The quantities similar to κ1 and κ2 in the geometry of surfaces are the principal curvatures, denoted λ1 , λ2 , λ3 . They are introduced as the eigenvalues of the so-called Weingarten linear map. The mean curvature at the point p is 1 H(p) = [λ1 (p) + λ2 (p) + λ3 (p)], 3 the Gauss-Kronecker curvature is K (p) = λ1 (p)λ2 (p)λ3 (p),

and the the scalar curvature is scal(p) = sec(σ1 ∧ σ2 ) + sec(σ2 ∧ σ3 ) + sec(σ3 ∧ σ1 ) = = λ1 λ2 + λ3 λ1 + λ2 λ3 . Bogdan D. Suceav˘ a California State University, Fullerton


Theorem Let M 3 ⊂ R4 be a smooth hypersurface and λ1 , λ2 , λ3 be its principal curvatures in the ambient space R4 endowed with the canonical metric. Let p ∈ M be an arbitrary point. Denote by H(p) = 13 (λ1 + λ2 + λ3 ) the mean curvature, by scal(p) = λ1 (p)λ2 (p) + λ3 (p)λ1 (p) + λ2 (p)λ3 (p) the scalar curvature at the point p ∈ M 3 . Then 1 H 2 (p) ≥ ρ(p) = scal(p), 3 with equality if and only if the point is umbilical, i.e. when λ1 (p) = λ2 (p) = λ3 (p).



Argument:



Argument: We have 9H 2 = (3H)2



Argument: We have 9H 2 = (3H)2 = (λ1 +λ2 +λ3 )2 =



Argument: We have 9H 2 = (3H)2 = (λ1 +λ2 +λ3 )2 = λ21 +λ22 +λ23 +2(λ1 λ2 +λ3 λ1 +λ2 λ3 ) ≥



Argument: We have 9H 2 = (3H)2 = (λ1 +λ2 +λ3 )2 = λ21 +λ22 +λ23 +2(λ1 λ2 +λ3 λ1 +λ2 λ3 ) ≥ ≥ 3(λ1 λ2 + λ3 λ1 + λ2 λ3 ). This last inequality holds true since λ21 + λ22 + λ23 ≥ λ1 λ2 + λ3 λ1 + λ2 λ3 , and in this inequality equality holds if and only if λ1 = λ2 = λ3 .



Argument: We have 9H 2 = (3H)2 = (λ1 +λ2 +λ3 )2 = λ21 +λ22 +λ23 +2(λ1 λ2 +λ3 λ1 +λ2 λ3 ) ≥ ≥ 3(λ1 λ2 + λ3 λ1 + λ2 λ3 ). This last inequality holds true since λ21 + λ22 + λ23 ≥ λ1 λ2 + λ3 λ1 + λ2 λ3 , and in this inequality equality holds if and only if λ1 = λ2 = λ3 . Since the scalar curvature of the hypersurface M 3 is scal = λ1 λ2 + λ3 λ1 + λ2 λ3 . This yields 9H 2 ≥ 3scal. The normalized scalar curvature in dimension three is ρ = 13 scal, therefore H 2 ≥ ρ, with equality if and only if the point is an umbilic. Bogdan D. Suceav˘ a California State University, Fullerton


Why is this important? Let M 3 ⊂ R4 be a smooth hypersurface and λ1 , λ2 , λ3 be its principal curvatures in R4 . Let p ∈ M be an arbitrary point. Denote by H(p) = 13 (λ1 + λ2 + λ3 ) the mean curvature, by scal(p) = sec(e1 ∧ e2 ) + sec(e3 ∧ e1 ) + sec(e2 ∧ e3 ) = λ1 (p)λ2 (p) + λ3 (p)λ1 (p) + λ2 (p)λ3 (p) the scalar curvature at the point p ∈ M 3 .



Why is this important? Let M 3 ⊂ R4 be a smooth hypersurface and λ1 , λ2 , λ3 be its principal curvatures in R4 . Let p ∈ M be an arbitrary point. Denote by H(p) = 13 (λ1 + λ2 + λ3 ) the mean curvature, by scal(p) = sec(e1 ∧ e2 ) + sec(e3 ∧ e1 ) + sec(e2 ∧ e3 ) = λ1 (p)λ2 (p) + λ3 (p)λ1 (p) + λ2 (p)λ3 (p) the scalar curvature at the point p ∈ M 3 . If scal(p) > 0, then the hypersurface can not be minimal.



Why is this important? Let M 3 ⊂ R4 be a smooth hypersurface and λ1 , λ2 , λ3 be its principal curvatures in R4 . Let p ∈ M be an arbitrary point. Denote by H(p) = 13 (λ1 + λ2 + λ3 ) the mean curvature, by scal(p) = sec(e1 ∧ e2 ) + sec(e3 ∧ e1 ) + sec(e2 ∧ e3 ) = λ1 (p)λ2 (p) + λ3 (p)λ1 (p) + λ2 (p)λ3 (p) the scalar curvature at the point p ∈ M 3 . If scal(p) > 0, then the hypersurface can not be minimal. Argument: H 2 ≥ 13 scal(p) > 0, hence H can not vanish at p.



This interpretation is related to John Nash’s Embedding Theorem.



This interpretation is related to John Nash’s Embedding Theorem. Problem: When does a given Riemannian manifold M admit (or does not admit) a minimal immersion into a Euclidean space of arbitrary dimension ?



Theorem (Bang-Yen Chen, 1993) Let M n be a submanifold in a space form of constant sectional curvature c. Then inf(sec) ≥ scal −

n2 (n − 2) 2 (n + 1)(n − 2) |H| − c. 2(n − 1) 2

The equality case is completely determined by the form of the shape operators with respect to a suitable o.n. frame fields.




n2 (n − 2) 2 (n + 1)(n − 2) |H| − c. 2(n − 1) 2

The equality case is completely determined by the form of the shape operators with respect to a suitable o.n. frame fields. Rewrite it in the form extrinsic ≥ intrinsic:




n2 (n − 2) 2 (n + 1)(n − 2) |H| − c. 2(n − 1) 2

The equality case is completely determined by the form of the shape operators with respect to a suitable o.n. frame fields. Rewrite it in the form extrinsic ≥ intrinsic: Theorem n2 (n − 2) 2 (n + 1)(n − 2) |H| + c ≥ scal − inf(sec). 2(n − 1) 2




n2 (n − 2) 2 (n + 1)(n − 2) |H| − c. 2(n − 1) 2

The equality case is completely determined by the form of the shape operators with respect to a suitable o.n. frame fields. Rewrite it in the form extrinsic ≥ intrinsic: Theorem n2 (n − 2) 2 (n + 1)(n − 2) |H| + c ≥ scal − inf(sec). 2(n − 1) 2 Chen’s curvature invariant δ = scal − inf(sec). If we study intrinsic curvature invariants, we actually pursue the program opened by Nash’s Embedding Theorem. Bogdan D. Suceav˘ a California State University, Fullerton


Theorem (Chen, 2005) Let M n be isometrically immersed in a Riemannian manifold ¯ n+m . Let sec, sec and scal(p) be the sectional curvature of M, M ¯ and the scalar curvature of M at p, the sectional curvature of M respectively. Then the following inequality holds: scal(p) ≤

n(n − 1) 2 X |H| + sec(ei ∧ ej ) 2 i