Timeline for Riemannian vs Non-Riemannian curvature

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Jan 5, 2018 at 13:59	comment	added	Robert Bryant		@AurelianoSkirzewski: No, your question is different from the one that I was answering in that other question because, as you stated, you only assumed that you were given $R$ as a $(1,3)$-tensor satisfying the first Bianchi identity (which is purely algebraic and does not involve a connection). Your question, as I understood it, was whether or not there was any metric $g$ for which $R$ was the Riemann curvature tensor. There is a separate, but weaker, question of whether or not $R$ is the curvature of some torsion-free connection, which it might be, even if there is no compatible metric $g$.
Jan 5, 2018 at 13:38	comment	added	Aureliano Skirzewski		I assumed there is a connection when I said it satisfies Bianchi identities. I focused on solving the problem of the algebraic restrictions on the curvature but my problem is with a connection. I thought of rephrasing to a question of whether a connection is Levi-Civita, but thinking in those terms I found an older question mathoverflow.net/questions/54434/… which you answered too. I believe the answer I was looking for may be in that other post. Thanks!
Jan 5, 2018 at 2:40	history	answered	Robert Bryant	CC BY-SA 3.0