The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group.
Is there a more general theorem that states that for some class of Lie groups or Riemannian manifolds (which includes the restricted Lorentz group), the exponential map is surjective?
There is a theorem stating that compact, connected Lie groups have surjective exponential maps. But as the restricted Lorentz group is not compact, this isn't applicable.
Answer
Comments to the question (v2):
The consensus in the literature seems to be that the surjectivity of the exponential map exp:so(1,d;R)→SO+(1,d;R) for the restricted Lorentz group for general spacetime dimensions D=d+1 does not have a short proof.
The case d=1 is trivial.
The case d=2 can be proved via the isomorphism SO+(1,2;R)≅SL(2,R)/Z2, cf. e.g. this Phys.SE post.
The case d=3 can be proved via the isomorphism SO+(1,3;R)≅SL(2,C)/Z2, cf. e.g. Wikipedia and this Phys.SE post.
Already the exponential map exp:sl(2,R)→SL(2,R) is not surjective, cf. e.g. this MO.SE answer and this Phys.SE post. Note that the Lie algebras so(1,2;R) ≅ sl(2,R) are isomorphic, but only the Lie group SO+(1,3;R) for the lefthand-side of the isomorphism (2) has a surjective exponential map; not the Lie group SL(2,R) for the righthand-side. A counterexample such as (2) undoubtedly makes it delicate to try to formulate a generalization of (1) beyond the restricted Lorentz groups SO+(1,d;R) and case-by-case-proofs. See also this Math.SE post.
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