On the first eigenvalue of non-orientable closed surfaces (Q1293157)

Let \( (M, g) \) be a 2-dimensional non-orientable closed Riemannian manifold and \( (\widetilde{M}, \widetilde{g}) \) its orientable Riemannian double cover. Let \(\lambda_1(M,g)\) (resp. \(\lambda_1(\widetilde{M},\widetilde{g})\)) denote the first nonzero eigenvalues of the Laplacian for functions on \((M,g)\) (resp. on \((\widetilde{M},\widetilde{g})\)). The author proves the following two results: (i) If \( M \) is homeomorphic to \( \mathbb{R} P ^2 , \) then \( \lambda _1 (M, g) > \lambda _1 ( \widetilde{M}, \widetilde{g}) \) for every metric \( g \) on \( M .\) (ii) If \( M \) is homeomorphic to \( \# ^n \mathbb{R} P ^2\) (the connected sum of \(n\) copies of the real projective plane) \((n \geq 2)\), there exists a metric \( g \) on \( M \) such that \( \lambda _1 (M, g) = \lambda _1 ( \widetilde{M}, \widetilde{g})\).