Continuous Groups of Transformation|Luther Pfahler Eisenhart

||boek: Continuous Groups of Transformations||Dover Books on Advanced Mathematics

||door: Luther Pfahler Eisenhart

||taal: en
||jaar: 1961
||druk: ?
||pag.: 301p
||opm.: paperback|like new

||isbn: N/A
||code: 1:002537

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Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

[source: https--www.amazon.com]

The classic 20th century text on the theory of continuous groups [2019-11-17]

This is a well written account of the classical approach to the theory of continuous groups by a leading expert from the early 20th century. The techniques used in the exposition are no longer current and so the exposition is not immediately transparent to a modern reader. Nevertheless, the text is useful and instructive for anyone interested in reading classical literature in mathematics. The time needed to unravel its content is time well invested and will reap ample rewards in deepening one's understanding of the theory. The text was a classic in the 20th century and was used by the majority of physicists in that era to learn this theory.

Hannibal Fourvector [source: https--www.amazon.com]

This book sets forth the general theory of Lie and his contemporaries and the results of recent investigations with the aid of the methods of the tensor calculus and concepts of the new differential geometry. The first three chapters contain in the main the results of the first period.

[source: https--www.e-booksdirectory.com/details.php?ebook=10403]

--- Over (foto 2): Luther Pfahler Eisenhart ---

Luther Pfahler Eisenhart (13 January 1876 - 28 October 1965) was an American mathematician, best known today for his contributions to semi-Riemannian geometry.

Life

Eisenhart was born in York, Pennsylvania, and graduated from Gettysburg College in 1896. He earned his doctorate in 1900 at Johns Hopkins University, where he was influenced (at long range) by the work of Gaston Darboux and at shorter range by that of Thomas Craig. During the next two decades, Eisenhart's research focused on moving frames after the French school, but around 1921 took a different turn when he became enamored of the mathematical challenges and entrancing beauty of a new theory of gravitation, Albert Einstein's general theory of relativity.

Eisenhart played a central role in American mathematics in the early twentieth century. He served as chairman of the mathematics department at Princeton University and later as Dean of the Graduate School there from 1933 to 1945. He is widely credited with guiding the development in America of the mathematical background needed for the further development of general relativity, through his influential textbooks and his personal interaction with Albert Einstein, Oswald Veblen, and John von Neumann at the nearby Institute for Advanced Study, as well as with gifted students such as Abraham Haskel Taub.

In the early 40s he chaired the "Reference Committee", formed in June 1940 for editors of scientific journals to send the papers submitted to them, in order to check that the papers did not contain results (especially regarding nuclear physics) whose public knowledge could be detrimental to the US war efforts.

Eisenhart was an elected member of both the American Philosophical Society and the United States National Academy of Sciences.

Publications

• Eisenhart, Luther Pfahler (1923). Transformations of Surfaces. Princeton: Princeton University Press. LCCN 62011699.
• Eisenhart, Luther Pfahler (1933). Continuous Groups of Transformations. Princeton: Princeton University Press. LCCN 61003361/L.
• Eisenhart, Luther Pfahler (1926). Riemannian Geometry (PDF). Princeton: Princeton University Press. OCLC 5836010.
• Eisenhart, Luther Pfahler (1939). Coordinate Geometry. Dover Publishing.
• Eisenhart, Luther Pfahler (1927). Non-Riemannian geometry (PDF). New York: American Mathematical Society.
• Eisenhart, Luther Pfahler (1909). A treatise on the differential geometry of curves and surfaces. Boston: Ginn and Company.

[source: wikipedia]

Luther Pfahler Eisenhart

Born: 13 January 1876 - York, Pennsylvania, USA
Died: 28 October 1965 - Princeton, New Jersey, USA
Summary: Luther Eisenhart was an American mathematician who worked in differential geometry and relativity theory.

Biography

Luther Eisenhart's parents were Charles Augustus Eisenhart and Emma Catherine Pfahler who were from long established York families. His father Charles Eisenhart had a surprisingly large number of different occupations as Eisenhart himself described (see [5]):-

"My father, after being a student at the York County Academy, taught in a country school until ... he went to Marshall, Michigan, and worked in a store. At the same time, he was apprentice to a local dentist. Being very expert with his hands, he soon acquired competence in the technique of dentistry. In due time he returned to York, set up a dentist office, and was married. He made sufficient income to meet the expenses of his growing family, but his intellect was too active to be satisfied by dentistry. Electricity appealed to him and he organized the Edison Electric Light Company in the early eighties. The telephone also made an appeal. He experimented with telephones and in the late nineties organized the York Telephone Company."

Luther was the second of his parents' six sons. He was brought up in a religious home with much of the family's social life being centred around St Paul's Lutheran Church. His mother provided him with such a good foundation to his education that, after beginning primary schooling at the age six and a half, he took only three years to complete the normal course of six years. He attended York High School but took the final year off school to prepare for entry to College undertaking independent study of Latin and Greek.

He was a student at Gettysburg College from September 1892 and won the prize for excellence in his first year and the mathematics prize in his second year. He also excelled at baseball, a sport that the young Eisenhart boys had been passionate about as they grew up. He had to spend the final two years at Gettysburg College studying mathematics with guidance from his professor but with no classes since he was the only student taking the subject. He was awarded his A.B. in 1896.

After teaching in the preparatory school of Gettysburg College for a year, he began graduate study at Johns Hopkins University in October 1897. He wrote [5]:-

"... Thomas Craig aroused my interest in differential geometry by his lectures and my readings of Darboux's treatises. Toward the close of 1900 I wrote a thesis in this field on a subject of my own choosing and in June the degree of Doctor of Philosophy was granted."

His thesis was entitled Infinitesimal deformation of surfaces. As Eisenhart indicated in the above quote, this work was heavily influenced by Darboux's treatise on the subject and he received little supervision for his doctorate.

Eisenhart spent most of his career at Princeton where he became an instructor in mathematics taking up his appointment in September 1900. In 1905 he was selected to be a preceptor, a position which had been recently created. In 1908 he married Anna Maria Dandridge Mitchell; they had one son Churchill Eisenhart. Sadly his wife died in 1913 and, five years later he married Katharine Riely Schmidt of York, Pennsylvania; they had two daughters. He was promoted to full professor in 1909 and worked at Princeton until he retired in 1945. He served as Dean of the Faculty from 1925 to 1933 when he became Dean of the Graduate School. After Henry Fine died following a bicycle accident in December 1928, Eisenhart became Head of Mathematics at Princeton and Dod Professor of Mathematics, continuing in these roles until he retired in 1945.

Before looking at his research contributions, let us look at some of the reforms he introduced at Princeton. He proposed the four course plan (instead of the five courses which was the scheme then in place) which involved the undergraduates doing independent reading and research during their final two years. They completed their course by writing an undergraduate thesis. The plan was adopted in 1923 and, as we can see from the details we gave above of Eisenhart's own education, it was very much based on the experiences he had gone through. He believed that:-

"... teaching methods . . . must be designed to encourage independence and self-reliance, to evoke curiosity, and stimulate the imagination and creative impulse."

He also stated that:-

"The real test of an educational process is what is becoming of the student as he proceeds with his education - how he is being prepared to continue his education and become an educated man."

There was opposition to the four course plan - both from fellow academics and from students. The students sang:-

"Luther Pfahler Eisenhart,
Efficient from the very start;
But he's condemned in the eyes of man
For originating the four-course plan."

However Eisenhart was prepared to fight for what he believed in, and his arguments won the day.

Let us now look at Eisenhart's research contributions. There are two stages in his work although it is all in differential geometry. The first stage continued his doctoral work studying deformations of surfaces. His first book A Treatise in the Differential Geometry of Curves and Surfaces , published in 1909, was on this topic and was a development of courses he had given at Princeton for several years. In [2] this book is described as:-

"... in textbook form, with numerous problems, introducing the student to classical and modern methods. One of the most interesting novelties of the volume was the so-called 'moving trihedrals' for twisted curves as well as surfaces so freely used in writings of Darboux and others. From the first, methods of the theory of functions of a real variable are employed. The work was of great value in introducing the American student to an important field by the most modern method of the time."

The second stage started after 1921 when Eisenhart, prompted by Einstein's general theory of relativity and the related geometries, studied generalisations of Riemannian geometry. He published Riemannian Geometry in 1926 and Non-Riemannian Geometry in 1927. The scene is set for the first of these works in [1]:-

"Riemann proposed the generalisation of the theory of surfaces as developed by Gauss, to spaces of any order, and introduced certain fundamental ideas in this general theory. Important contributions to it were made by Bianchi, Beltrami, Christoffel, Schur, Voss, and others, and Ricci-Curbastro coordinated and extended the theory with the use of tensor analysis and his absolute calculus. The book gave a presentation of the existing theory of Riemannian geometry after a period of considerable study and development of the subject by Levi-Civita, Eisenhart, and many others."

In 1933 Eisenhart published Continuous Groups of Transformations which continues the work of his earlier books looking at Lie's theory using the methods of the tensor calculus and differential geometry. Again quoting [1]:-

"The study of continuous groups of transformations inaugurated by Lie resulted in the developments by Engel, Killing, Scheffers, Schur, Cartan, Bianchi and Fubini, a chapter which closed about the turn of the century. The new chapter began about 1920 with the extended studies of tensor analysis, Riemannian geometry and its generalizations, and the application of the theory of continuous groups to the new physical theories. Eisenhart has thus developed a remarkable body of original material and has notably served his colleagues by frequent surveys of fields in which he had become a specialist."

After he retired, Eisenhart continued to undertake research. In fact he published 21 papers between 1951 and 1963, for example: Generalized Riemann spaces and general relativity (1953); A unified theory of general relativity of gravitation and electromagnetism (1956); The cosmology problem in general relativity (1960); and The Einstein generalized Riemannian geometry (1963).

Eisenhart had a long association with the American Mathematical Society being vice president in 1914, and Colloquium lecturer in 1925 when he lectured on non-Riemannian geometry. He edited The Annals of Mathematics from 1911 to 1925 and the Transactions of the American Mathematical Society from 1917 to 1923, being managing editor in 1920-23. He was President of the American Mathematical Society from 1931 to 1932. He was also honoured by being elected president of the American Association of Colleges (1930), and vice president of the National Academy of Sciences (United States) (1945-49) and of the American Association for the Advancement of Science. He was also elected an officer of the American Philosophical Society serving from 1942 to 1959. In addition he received honorary degrees from Gettysburg College (1921), Columbia University (1931), University of Pennsylvania (1933), Lehigh University (1935), Duke University (1940), Princeton University (1952) and Johns Hopkins University (1953). In 1937 he was named Officer of the Order of the Crown of Belgium.

Lefschetz writes about Eisenhart's character in [1]:-

"He was par excellence a family man and found in his family a great source of happiness and strength. Eisenhart was essentially a most modest man. The intimate atmosphere which surrounded him, its very serenity, was due in large measure to the care and devotion which he received from Mrs Eisenhart. The Dean, as we all called him, did not seem to realize that he was an outstanding leader both in his field and in higher education. For outside his family he had two "loves": differential geometry (as research and study) and education."

After his death his faculty colleagues said, in a tribute, that:-

"... in two centuries of Princeton's history few scholars had done more to shape the future of the University ..."

and the trustees declared that he had:-

"... earned an enduring place in the front rank [of those who had] made Princeton great."

[source: https--mathshistory.st-andrews.ac.uk/Biographies/Eisenhart]

Luther Pfahler Eisenhart

President, 1931-1932

Ph.D. Johns Hopkins University, Baltimore, Maryland, 1900

Eisenhart published a wealth of research in the field of differential geometry and its physical applications. He developed a remarkable body of original work and served his colleagues by publishing surveys of fields in which he had become a specialist. He was the first American to publish a general work in the field of differential geometry, for instance.

Eisenhart spent most of his career at Princeton University (1900 until his retirement in 1945); he became professor in 1909, served as dean of the faculty from 1923 to 1933, and was appointed dean of the graduate school in 1933. Over the years Eisenhart had a varied association with the AMS, serving as Vice President, editor of Transactions of the American Mathematical Society, Colloquium Lecturer, and President. Eisenhart was a member of the U.S. National Academy of Sciences.

[source: http--www.ams.org/about-us/presidents/21-eisenhart]