1. Overview
Elwin Bruno Christoffel (Elwin Bruno ChristoffelkʁɪˈstɔflGerman; November 10, 1829 - March 15, 1900) was a prominent German mathematician and physicist. He is widely recognized for his foundational contributions to differential geometry, which were instrumental in paving the way for the development of tensor calculus. This mathematical framework later became the essential basis for Einstein's general theory of relativity. Beyond differential geometry, Christoffel also made significant advancements in complex analysis, particularly with the Schwarz-Christoffel mapping, and in numerical analysis, where he generalized Gaussian quadrature and developed the Christoffel-Darboux formula. His diverse research also extended into potential theory, differential equations, the pioneering study of shock waves, and optics.
2. Life
Elwin Bruno Christoffel's life was marked by a deep commitment to academic pursuits, from his early education in his hometown to his influential professorships at various prestigious European universities.
2.1. Early Life and Education
Christoffel was born on November 10, 1829, in Montjoie (now known as Monschau), which was then part of Prussia. His family were successful cloth merchants. He received his initial education at home, focusing on languages and mathematics. Following this, he attended the Jesuit Gymnasium and the Friedrich-Wilhelms Gymnasium, both located in Cologne. In 1850, he enrolled at the Humboldt University of Berlin, where he pursued studies in mathematics, physics, and chemistry. During his time in Berlin, he was particularly influenced by the renowned mathematician Gustav Dirichlet. Christoffel completed his doctoral studies at the University of Berlin in 1856. His thesis focused on the motion of electricity in homogeneous bodies, and he conducted this research under the guidance of Martin Ohm, Ernst Kummer, and Heinrich Gustav Magnus.
2.2. Academic Career
After earning his doctorate, Christoffel returned to Montjoie, where he spent three years largely isolated from the broader academic community. Despite this isolation, he continued his intensive study of mathematics, particularly mathematical physics, drawing from the works of prominent mathematicians such as Bernhard Riemann, Dirichlet, and Augustin-Louis Cauchy. During this period, he also continued his own research, publishing two significant papers in differential geometry.
In 1859, Christoffel returned to Berlin, where he completed his habilitation and became a PrivatdozentGerman (private lecturer) at the University of Berlin. In 1862, he was appointed to a professorial chair at the Polytechnic School in Zürich, a position that had been vacated by Richard Dedekind. At this relatively young institution, established only seven years prior, Christoffel played a crucial role in organizing a new and highly regarded institute of mathematics. His research continued to flourish, and in 1868, he was recognized for his contributions by being elected as a corresponding member of both the Prussian Academy of Sciences and the Istituto Lombardo in Milan.
In 1869, Christoffel moved back to Berlin to take up a professorship at the Gewerbeakademie, an institution that is now part of the Technische Universität Berlin. His former position in Zürich was filled by Hermann Schwarz. However, the Gewerbeakademie faced stiff competition from the nearby and well-established University of Berlin, making it challenging to attract a sufficient number of students for advanced mathematics courses. Consequently, Christoffel left Berlin again after three years.
In 1872, Christoffel accepted a professorship at the University of Strasbourg. This ancient institution was undergoing a significant reorganization into a modern university following Prussia's annexation of Alsace-Lorraine after the Franco-Prussian War. Alongside his colleague Theodor Reye, Christoffel was instrumental in building a highly reputable mathematics department at Strasbourg. He continued to be a prolific researcher and mentored several doctoral students, including Rikitaro Fujisawa, Ludwig Maurer, and Paul Epstein.
2.3. Retirement and Death
Christoffel retired from the University of Strasbourg in 1894, and his position was taken over by Heinrich Martin Weber. Even after retirement, he remained actively engaged in his work, continuing to conduct research and publish. His final treatise was completed just before his death and was published posthumously. Elwin Bruno Christoffel died on March 15, 1900, in Strasbourg. He remained unmarried throughout his life and left no immediate family.
3. Major Contributions
Elwin Bruno Christoffel's scientific and mathematical achievements spanned several fields, leaving a lasting impact on the development of modern mathematics and physics. His work encompassed conformal mapping, potential theory, invariant theory, tensor analysis, mathematical physics, geodesy, and the study of shock waves.
3.1. Differential Geometry
Christoffel is most renowned for his groundbreaking contributions to differential geometry. In a highly influential paper published in Crelle's Journal in 1869, which addressed the equivalence problem for differential forms in *n* variables, he introduced the fundamental technique now known as covariant differentiation. He utilized this concept to define the Riemann-Christoffel tensor, which remains the most common method for expressing the curvature of Riemannian manifolds. In the same paper, he also introduced the Christoffel symbols (Γkij and Γkij), which represent the components of the Levi-Civita connection with respect to a system of local coordinates.
Christoffel's innovative ideas were subsequently generalized and significantly expanded upon by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita. They transformed these concepts into the broader framework of tensors and the absolute differential calculus. This absolute differential calculus, later renamed tensor calculus, ultimately formed the essential mathematical foundation for Einstein's general theory of relativity.
3.2. Complex Analysis
Christoffel also made important contributions to complex analysis. His work includes the Schwarz-Christoffel mapping, which was the first non-trivial and constructive application of the Riemann mapping theorem. This mapping has found numerous applications in the theory of elliptic functions and various areas of physics. In the field of elliptic functions, he also published significant results concerning abelian integrals and theta functions. The Schwarz-Christoffel mapping was published by Christoffel in 1867.
3.3. Numerical Analysis
In numerical analysis, Christoffel generalized the Gaussian quadrature method for integration. In connection with this work, he introduced the Christoffel-Darboux formula for Legendre polynomials in 1858. He later extended this formula to apply to general orthogonal polynomials.
3.4. Mathematical Physics and Other Research
Christoffel's research extended into various other domains, including potential theory and the theory of differential equations. Although much of his work in these areas went largely unnoticed during his time, he did publish two pioneering papers on the propagation of discontinuities in the solutions of partial differential equations. This work represented an early and significant contribution to the theory of shock waves. He also engaged in physics research, publishing work in optics. However, his contributions in optics quickly lost their relevance with the eventual abandonment of the concept of the luminiferous aether. His broader work also touched upon invariant theory and geodesy.
4. Personal Life
Elwin Bruno Christoffel maintained a private personal life. He never married and had no family.
5. Honours and Memberships
Christoffel's significant contributions to mathematics and physics were recognized through his election as a corresponding member to several prestigious academic academies:
- Prussian Academy of Sciences (1868)
- Istituto Lombardo (1868)
- Göttingen Academy of Sciences (1869)
Additionally, the Kingdom of Prussia awarded him two distinctions for his academic and scientific activities:
- Order of the Red Eagle 3rd Class with bow (Schleife) (1893)
- Order of the Crown 2nd Class (1895)
6. Legacy and Evaluation
Elwin Bruno Christoffel's work left an indelible mark on the fields of mathematics and physics, particularly through his foundational contributions to differential geometry.
6.1. Influence on Later Developments
Christoffel's introduction of covariant differentiation, the Riemann-Christoffel tensor, and the Christoffel symbols provided the essential groundwork for the development of tensor calculus. This powerful mathematical tool was subsequently adopted and extensively developed by mathematicians such as Gregorio Ricci-Curbastro and Tullio Levi-Civita. The absolute differential calculus, which evolved into tensor calculus, proved to be indispensable, forming the rigorous mathematical basis for Einstein's revolutionary general theory of relativity. Without Christoffel's early insights into the geometry of curved spaces, the mathematical formulation of general relativity would have been significantly more challenging.
6.2. Scholarly Assessment
Within the history of mathematics, Christoffel's achievements are considered fundamental. His work on the Riemann curvature tensor and Christoffel symbols provided the necessary machinery to describe intrinsic curvature, a concept crucial for understanding the geometry of manifolds. His contributions to complex analysis, particularly the Schwarz-Christoffel mapping, demonstrated a powerful constructive application of the Riemann mapping theorem and found practical uses in various scientific disciplines. Furthermore, his generalizations in numerical analysis, such as the Gaussian quadrature method and the Christoffel-Darboux formula, continue to be relevant in computational mathematics. Overall, Christoffel's rigorous and innovative approach to mathematical problems laid critical groundwork that profoundly impacted subsequent scientific progress, particularly in theoretical physics.