How Did Riemann Arrive at His Curvature?

The Philadelphia Area Seminar on History of Mathematics will meet on Thursday, March 21, 2019,
 
from 6:00 p.m. to 8:00 p.m. in Saint Augustine Center, Room 300.  (Parking Permits to follow.)
 
We begin with conversation and a light supper (donation $10.00). About 6:30-6:45 the talk will begin.

With one lecture delivered to the philosophical faculty of the University of Göttingen in 1854, Bernhard Riemann created the rich subject that we now call Riemannian geometry.  Since most of the audience were not mathematicians, his lecture described results and their implications without many formulas or calculations and with only sketches of arguments. This, and the paucity of related documents, makes it difficult to see how Riemann arrived at his results.  To reconstruct the development of Riemann’s ideas, therefore, historians have had to resort to speculation.  Gauss’s work on the intrinsic geometry of a surface certainly inspired Riemann, but historians have suggested varied explanations of the leap from the Gaussian curvature of a surface to the Riemannian curvature of an n-dimensional manifold.  I shall briefly touch on some of those explanations and then offer a reconstruction of my own that may at least supplement these other explanations.
 
Directions:  Villanova University is located on Route 30, Lancaster Avenue, just east of I-476.  If you drive to the meeting, enter Villanova by the main gate on Ithan Avenue. By order of the Parking Office all parking is now in the Saint Augustine Center lot, newly refurbished to a multistory structure. You will need to display a parking pass, which will follow in a later email. The Saint Augustine Center is attached to the lot, and Room 300 is on the third floor. You can get in touch with me (Alan Gluchoff) at 610-905-1376 if you need help.
 
Public Transportation: Take SEPTA’s Paoli-Thorndale train to the Villanova station. If outbound from center city go down through the tunnel to the inbound side. From the inbound platform go up the stairs and proceed to the left.  The second building you encounter is the SAC; entry is through the parking lot or the front entrance.