  We prove that the set of orthogonal projections on a Hilbert space equipped
with the length metric is $\frac\pi2$-geodesic. As an application, we consider
the problem of variation of spectral subspaces for bounded linear self-adjoint
operators and obtain a new estimate on the norm of the difference of two
spectral projections associated with isolated parts of the spectrum of the
perturbed and unpertubed operators, respectively. In particular, recent results
by Kostrykin, Makarov and Motovilov from [Trans. Amer. Math. Soc., V. 359, No.
1, 77 -- 89] and [Proc. Amer. Math. Soc., 131, 3469 -- 3476] are sharpened.
