One and two (space-time) dimensional general relativity

This is a bit of a silly post, partly written because I came across this paper when doing some literature search.

To give a spoiler: the punchline is

Life is very boring in one and two space-time dimensions.

Readers familiar with Einstein’s equations (with cosmological constant) may want to pause and think for a few seconds; then they may want to skip reading this post entirely.

Mariş’s Theorem

During a literature search (to answer my question concerning symmetries of “ground states” in variational problem, I came across a very nice theorem due to Mihai Mariş. The theorem itself is, more than anything else, a statement about the geometry of Euclidean spaces. I will give a rather elementary write-up of (a special case of) the theorem here. (The proof presented here can equally well be applied to get the full strength of the theorem as presented in Maris’s paper; I give just the special case for clarity of the discussion.)