latentcorpse
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define the group
S0_O(3,1) = \{ a \in SO(3,1) | (ae_4,e_4) < 0 \}
i need to show this group is connected
in my notes it says a group G is connected if it satisfies any of the following:
(i)any two elements of G can be joined by a C^k-path in G
(ii) it is not the disjoint union of two non-empty open sets
(iii) it is generated by a neighbourhood of 1 (the identity matrix)
(iv) it is generated by exp \mathfrak{g}
im not sure which one to try and prove or how to go about it really. can anybody offer some advice? thanks.
S0_O(3,1) = \{ a \in SO(3,1) | (ae_4,e_4) < 0 \}
i need to show this group is connected
in my notes it says a group G is connected if it satisfies any of the following:
(i)any two elements of G can be joined by a C^k-path in G
(ii) it is not the disjoint union of two non-empty open sets
(iii) it is generated by a neighbourhood of 1 (the identity matrix)
(iv) it is generated by exp \mathfrak{g}
im not sure which one to try and prove or how to go about it really. can anybody offer some advice? thanks.