mathematics

Stefan Cohn-Vossen

Stefan Cohn-Vossen is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Stefan Cohn-Vossen rather than just read about it. In short: Stefan Cohn-Vossen (28 May 1902 – 25 June 1936) was a mathematician, specializing in differential geometry. He is best known for his collaboration with David Hilbert on the 1932 book Anschauliche Geometrie, translated into English as Geometry and the Imagination.

Key takeaways

Stefan Cohn-Vossen belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Stefan Cohn-Vossen to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Stefan Cohn-Vossen from memory before moving on to harder problems.

Reference excerpt

Stefan Cohn-Vossen (28 May 1902 – 25 June 1936) was a mathematician, specializing in differential geometry. He is best known for his collaboration with David Hilbert on the 1932 book Anschauliche Geometrie, translated into English as Geometry and the Imagination. Both Cohn-Vossen's inequality and the Cohn-Vossen transformation are named after him. He also proved the first version of the splitting theorem.

Biography Stefan Cohn-Vossen was born 28 May, 1902 to Emanuel Cohn, a lawyer, and Hedwig (née Vossen) in Breslau (then a city in the Kingdom of Prussia; now Wrocław in Poland). He attended Göttingen in 1920; his notes from Hilbert's lectures on geometry at that time would form the basis for Anschauliche Geometrie. He wrote a 1924 doctoral dissertation at the University of Breslau (now the University of Wrocław) under the supervision of Adolf Kneser. In 1929 he completed his habilitation at Göttingen with his thesis Non-rigid closed surfaces under Richard Courant. Cohn-Vossen became a professor at the University of Cologne in 1930. In 1931, Cohn-Vossen married Dr. Margot Maria Elfriede Ranft. Anschauliche Geometrie was published in 1932; the book was well reviewed and by association with Hilbert, Cohn-Vossen became well known. He was barred from lecturing in 1933 under Nazi racial legislation, because he was Jewish. Unable to work in Germany, Cohn-Vossen moved to Switzerland in 1934, first to Locarno and then to Zurich, where he taught gymnasium. His son, Richard Cohn-Vossen, was born in Zurich in September 1934. Both Courant and Karl Löwner recommended Cohn-Vossen for positions abroad, at Istanbul and Dartmouth respectively; ultimately he emigrated to the USSR, with support from Herman Müntz, Fritz Houtermans, Pavel Alexandrov, and Heinz Hopf, where he was appointed to the Academy of Sciences and worked at Leningrad State University and the Steklov Institute. Cohn-Vossen died in Moscow from pneumonia in 1936. Despite his short time in the USSR, Cohn-Vossen had a significant impact on the development of differential geometry "in the large" in Soviet mathematics. Following Cohn-Vossen's death, his widow, Dr. Elfriede Cohn-Vossen, remarried Alfred Kurella, returned to Germany in 1954, and died in 1957. His son, Richard, became a filmmaker. In 1946, unaware of his death, the University of Cologne offered Cohn-Vossen his professorship back.

Publications

Books Hilbert, David; Cohn-Vossen, Stefan (1932). Anschauliche Geometrie [Geometry and the Imagination] (in German). Berlin: Springer. Cohn-Vossen, S. E. (Кон-Фоссен С. Э.) (1959). Некоторые вопросы дифференциальной геометрии в целом [Some Problems of Differential Geometry in the Large] (in Russian). Moscow: Fizmatgiz.

Articles Cohn-Vossen, Stephan (1927). "Singularitäten konvexer Flächen" [Singularities of convex surfaces]. Math. Ann. (in German). 97: 377–386. doi:10.1007/BF01447873. Cohn-Vossen, Stefan (1927). "Zwei Sätze über die Starrheit der Eiflachen" [Two theorems on the rigidity of the ellipsoid]. Nach. Gesellschaft Wiss. Gottingen, Math. Phys. Kl. (in German): 125–134. Cohn-Vossen, Stefan (1928). "Die parabolische Kurve. Beitrag zur Geometrie der Berührungstransformationen, der partiellen Differentialgleichung zweiter Ordnung und der Flächenverbiegung". Math. Ann. 99: 273–308. doi:10.1007/BF01459097. Cohn-Vossen, Stefan (1930). "Unstarre geschlossene Flächen" [Non-rigid closed surfaces]. Math. Ann. (in German). 102: 10–29. doi:10.1007/BF01782336. Cohn-Vossen, Stefan (1933). "Sur la courbure totale des surfaces ouvertes". C. R. Acad. Sci. Paris (in French). 197: 1165–1167. Cohn-Vossen, Stefan (1935). "Kürzeste Wege und Totalkrümmung auf Flächen" [Shortest paths and total curvature on surfaces]. Compositio Mathematica (in German). 2: 69–133. Cohn-Vossen, Stefan (1935). "Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken" [Total curvature and geodesics on simply connected open patches]. Mat. Sbornik (in German). 43 (2): 139–164. Cohn-Vossen, Stefan (1936). "Der approximative Sinussatz für kleine Dreiecke auf krummen Flächen (Auszug aus einem Brief an Prof. T. Levi-Civita)" [The approximate sine rule for small triangles on curved surfaces]. Compositio Mathematica (in German). 3: 52–54. Cohn-Vossen, Stefan (1936). "Existenz kürzester Wege" [The existence of shortest paths]. Compositio Mathematica (in German). 3: 441–452. Cohn-Vossen, Stephan (1938). "Die Kollineationen desn-dimensionalen Raumes" [Collineations of n-dimensional space]. Math. Ann. (in German). 115: 80–86. doi:10.1007/BF01448928.

Illustrations

Worked examples

Example 1 — a first encounter with Stefan Cohn-Vossen

Start with the simplest possible case. Write down what Stefan Cohn-Vossen claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Stefan Cohn-Vossen before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Stefan Cohn-Vossen ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Stefan Cohn-Vossen

In research: Stefan Cohn-Vossen appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Stefan Cohn-Vossen in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Stefan Cohn-Vossen is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1902 births, 1936 deaths, 20th-century German mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Stefan Cohn-Vossen outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Stefan Cohn-Vossen in 20 minutes

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Frequently asked questions

What is Stefan Cohn-Vossen in simple terms?

Why does Stefan Cohn-Vossen matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Stefan Cohn-Vossen?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Stefan Cohn-Vossen.