I've seen several posts tring to generalize the fact that the first eigenfunction of laplacian is the only nonvanishing eigenfuction, but I've no idea how to prove that fact nor I could find any reference of giving a proof. I guess the argument should be simple, but I just can't come up with it.
I'll really appreciate if anyone can give me good references or simply sketch the idea of how to prove that fact. Thanks in advance!