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OP's 1st (2nd) action is (is not) covariant/geometric, respectively. OP's 2nd action transforms with a boundary term under general coordinate transformations. The reason for the GHY boundary term in the first place is explained e.g. in this Phys.SE post. OP asks in a comment: Why is GHY term needed? Isn't space infinite without a boundary? When deriving ...


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