I have seen suggested that someone other than Riemann first came up with the Riemann Sphere. Is this correct? And if so, who did invent it?
After some searching I've found the following. Carl G. Neumann is given credit for inventing the Riemann Sphere. Also, one source gives Felix Klein independently -- Jeremy Gray in Worlds Out of Nothing: A course in the history of geometry in the 19th century, p. 344. Gray notes that "Klein observed [in a paper of 1874] that Riemann's treatment of what happened out towards infinity had the effect of making infinity a point, and that this could be seen by stereographic projection. We might add that, topologically, this is the one-point compactification of the plane." Gray goes on to say that Klein was uncertain what to make of this because the idea conflicted with what was currently accepted in projective geometry.
Now on to Neumann. Interestingly, the well known geometer H. S. M. Coxeter refers to the sphere as the Neumann Sphere on p. 258 of Section 14.6 in Non-Euclidean Geometry (1998) 6th edition. He might be referencing earlier books such as Theory of functions of a complex variable by Heinrich Burkhardt and Samuel Eugene Rasor (1913) which states on p. 47
"I. Place a sphere* of unit diameter on the xy plane (considered horizontal) so that it touches the plane at the origin O. The highest point of the sphere -- that one which lies diametrically opposite to O -- will be called O'. From this point O', project the points of the plane on the sphere by straight lines. This kind of projection has been used since the earliest times in cartography under the name of stereographic projection.
*This sphere is called Neumann's sphere. In following out one of Riemann's ideas Neumann chose the sphere instead of the plane as the field of the complex variable. It is used by Neumann throughout his treatise Vorlesungen Uber Riemann's Theorie der Abel'schen Integrale (Leipzig Teubner 2d ed 1884).-- SER"
Several other books also refer to Neumann as first using what we call the Riemann sphere.
1) Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Umberto Bottazzini, Jeremy Gray. (2013). pp. 278, 701.
2) Classical Complex Analysis. Mario Gonzalez. (1991). p 59.
3) A History of Geometrical Methods. Julian Lowell Coolidge. Courier Dover Publications, 2003. p. 215.
4) The Foundations of Geometry and the Non-Euclidean Plane. Martin, G. E. (1975). Springer Science & Business Media. P. 312.