surfing the internet to look for good books and have stumbled on this one, just scanning few pages i can already say this is a good book, may come handy for pre-requisite refresher topics
Introduction to Differential Geometry
https://link.springer.com/book/10.1007/978-3-662-64340-2
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(just posting today, an mwfsat day, what has been done yesterday, a tthsun day), here, adhering to what is analogous to Kaluza's cylinder hypothesis
https://foxtrotpaparomeo.wordpress.com/2023/01/11/11012023rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/
Are these equations extensible to a 3+1 spacetime?
Are you thinking about this or are you ghosting me?
polishing it with some corrections, i might exceed five pages
https://foxtrotpaparomeo.wordpress.com/2023/03/20/20032023rough-draftstatic-solutions-to-einsteins-field-equations-in-11-space-time/