A method of mapping a regular-range cylindrical image to a spherical surface in an ios application and displaying it on a screen with a spherical body.

I think it would be easier to get an answer if there were some explanations about what level of response the questioner is looking for and what kind of app configuration they are trying to implement.(Maybe it's because there's no explanation for not being able to answer or comment after a week?)
The term "level" here is similar to the term "level" used to categorize programming languages into low-level (low-level) and high-level (high-level) languages.If it's a low level answer, it's Let's study OpenGLES.OpenGL is a language or device independent 3D graphics API, while Mac OS, iOS and others use OpenGL ES for 3D graphics technology (ES refers to a slightly easier API in OpenGL).The GLKit Framework is provided as a means of implementing OpenGLES within the program.

iOS OpenGLES Programming Guide (PDF Documentation)

If you consider mapping to a sphere as a kind of photo retouching, such as blurring or color adjustment, you may want to check out the CoreImage Framework.You will find that the CIFilter class allows you to process different images.

Core Image Programming Guide

However, there is (probably) no "spherical mapping" filter in the genuine Apple filter.

The highest level of response is to use SceneKit.You can edit 3D graphics without needing (almost) knowledge of OpenGL, which is certainly the best answer.

Mapping Images

Results of mapping using "normal range cylindrical drawing"

However, the downside of this response is that SceneKit is a framework for creating games, not purely 3D graphics.If it's graphics, you want it to be printed as a file or NSData object in the end, but I can't do that (it seems that UIImage can take it out).