I'm using Inkscape to draw my diagrams and I would like to create a diagram of a singularity, so that it looks like a disk that is bent downwards in the middle, so a reverse tent, kind of like the diagrams used to illustrate the gravity wells of black holes. I was hoping I could convert an ellipse to a path, then just insert a node in the middle, drag the whole thing downwards and then add some shading, but my attempts to do this have not been successful. Is there a way to do this?
Seemingly you hope that an ellipse is actually a circle, only seen from other direction than perpendicularly. But it's not. In Inkscape it's only a 2D curve in XY plane and Inkscape has no 3D space around it. What you wanted is literally possible in advanced surface modelling 3D programs which allow freeform surfaces that can be modified like one modifies paths in Inkscape, Illustrator etc... 2D drawing programs.
In 3D programs you'll get also proper shading as a free extra without any effort after the surface geometry is defined.
Moving a node downwards to stretch a surface is not the only possible way to get what you want in 3D programs. You can also revolve a curve around a vertical line. Revolving for example a part of curve Y=1/X or only something resembling around the Z-axis may well be good enough.
Then there's math plotting programs. Check gnuplot. It's free. Get it from here http://www.gnuplot.info/
You must be able to input what you want as programming expressions, but probably that's no problem in your case. The plotted image can be exported as vector graphics which can be used in Inkscape.
Some websites make math plots - no need to install anything. Check for eample this https://c3d.libretexts.org/CalcPlot3D/index.html
The next screenshot is from there:
The site looks simple, but there are numerous settings for variations and the image can be printed as PDF which is a vector.