# Why a sphere cannot have a Lorentzian Metric?

I was listening to a Lecture and the lecturer said that a sphere cannot have a Lorentzian Metric. Is that accurate? If so, why?

• Possible duplicate: mathoverflow.net/questions/44861/… – lemon Mar 8 '16 at 9:54
• Did he mean a two-dimensional sphere? If so, it's because the light cone could be used to define a non-vanishing vector field, contradicting the Hairy Ball Theorem. – PseudoNeo Mar 8 '16 at 9:55
• I'm not sure what he meant, was just a one sentence statement. If the reason is what you're saying, then nothing is implied on $S^3$ – Dac0 Mar 8 '16 at 10:08
• S³ does have a Lorentzian metric, given the answers of mathoverflow.net/questions/44861/… – PseudoNeo Mar 8 '16 at 10:15
• It seems his statement was simply incorrect – Dac0 Mar 8 '16 at 10:21