First of all, I would like to state that my background is physics, and I apologize for possibly using erroneous definitions (names) in this question, as I have never been trained in topology.
I have been reading about pseudospheres and studying their properties for a while. I managed to calculate some of their properties that I was interested in (volume, surface area, Gaussian curvature etc.). Then, I thought about the generalization of pseudospheres to higher dimensions. It is relatively straightforward to do it for spheres and there are lots of resources around. However, I could not find much for pseudospheres. I encountered this question (The pseudosphere in 2 and in n dimensions.) where the commenter suggested that n-pseudosphere can be embedded in hyperbolic (2n-1)-space. I checked the reference that they provided ( Tenenblat and Terng, Bäcklund's Theorem for n-Dimensional Submanifolds of ℝ2−1 R 2 n − 1 , Annals of Mathematics, 111 (1980), pp. 477-490), and found that the generalization involves rotating the pseudosphere around another axis, and it is claimed that they cannot be embedded in less than 2n-1 space.
However, I found another paper (Tom M. Apostol & Mamikon A. Mnatsakanian (2015) Volume/Surface Area Relations for n-Dimensional Spheres, Pseudospheres, and Catenoids, The American Mathematical Monthly, 122:8, 745-756) in which the authors, as far as I understand, embedded pseudospheres in n+1 dimensional space. They also claimed that surface area of an n-sphere is equal to that of n-pseudosphere (true in 3D).
Are these two papers in conflict? Or, am I missing something crucial in this topic? I would appreciate any help.
Thank you.