What were Riemann's other two submissions for his habilitation?

Question

In Stalking the Riemann Hypothesis, Rockmore discusses how Bernhard Riemann, as per custom, submitted three potential areas of research for his habilitation. Gauss was the chairman of the committee responsible for picking which of the three Riemann would work on, and, perhaps because of his interest in non-Euclidean geometry, he selected the third one, regarding the "foundations of geometry". Riemann's work there would, or course, lead to the development of Riemannian geometry.

What were Riemann's other two submissions? Did he go on later in life to research those topics?

Answer 1

See :

• Detlef Laugwitz, Bernhard Riemann 1826-1866 : Turning Points in the Conception of Mathematics (1996 - German ed.1996), page 182; we have :

Riemann's habilitation paper "Uber die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe" (W.227-265) ("On the representability of a function by means of a trigonometric series") was completed in 1853, but was published by Dedekind only in 1868.

and page 219 :

Riemann [...] habilitation lecture "Uber die Hypothesen die der Geometrie zu Grunde liegen" ("On the hypotheses which lie at the foundation of geometry") (W.272-287), presented in the summer of 1854 [also published by Dedekind in 1868].

As is done even today, Riemann was supposed to submit titles of three lectures, from which the faculty of philosophy was to choose one. The first title was "Geschichte der Frage uber die Darstellbarkeit einer Function durch eine trigonometrische Reihe" ("The history of the question of representability of a function as a trigonometric series"), and the second, "Uber die Auflosung zweier Gleichungen zweiten Grades mit zwei unbekannten Grossen" ("Solution of two quadratic equations in two unknowns"). Of course, the third title was "Uber die Hypothesen..."