Integral geometry of real surfaces in complex projective spaces (Q5949481)

A classical result in integral geometry is the Poincaré formula for the average of the intersection number of two curves. Many extensions of this result are found in the literature and the present authors provide one particular generalization related, too, to the article by \textit{R. Howard} [The kinematic formula in Riemannian homogeneous spaces, Mem. Am. Math. Soc. 509 (1993; Zbl 0810.53057)]. The main result in the article is the following Theorem. Let \(\mathbb{C} \mathbb{P}^n\) be a complex projective space of complex dimension \(n\), \(M\) a real submanifold of \(\mathbb{C}\mathbb{P}^n\) of real dimension 2 and \(N\) a complex submanifold of complex dimension \(n-1\). Then we have NEWLINE\[NEWLINE\int_{U(n+1)}\#(M\cap gN)d\mu_{U(n+ 1)}(g)= {\text{vol} \bigl(U(n+1)\bigr) \text{vol} (N)\over 2 \text{vol} (\mathbb{C} \mathbb{P}^1) \text{vol} (\mathbb{C}\mathbb{P}^{n-1})} \int_M(1+ \cos^2 \theta_x)d \mu_M(x)NEWLINE\]NEWLINE where \(\theta_x\) is the Kähler angle of \(M\) at \(x\). Moreover the above formula holds for a real submanifold \(M\) of real dimension \(2n-2\) and a complex submanifold \(N\) of complex dimension 1, where \(\theta_x\) is the Kähler angle of \(T_x^\perp M\).