The $ER\left( 2 \right)$-cohomology of $B\mathbb{Z}/\left( {{2}^{q}} \right)$ and $\mathbb{C}{{\mathbb{P}}^{n}}$ are computed along with the Atiyah–Hirzebruch spectral sequence for $ER{{\left( 2 \right)}^{*}}\left( \mathbb{C}{{\mathbb{P}}^{\infty }} \right)$. This, along with other papers in this series, gives us the $ER\left( 2 \right)$-cohomology of all Eilenberg–MacLane spaces.

We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$-parallel and satisfies a further condition.

We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

We give a characterization of a minimal real hypersurface with respect to the condition for the sectional curvature.

Starting from two Lagrangian immersions and a horizontal curve in S3(1), it is possible to construct a new Lagrangian immersion, which we call a warped-product Lagrangian immersion. In this paper, we find two characterizations of warped-product Lagrangian immersions. We also investigate Lagrangian submanifolds which attain at every point equality in the improved version of Chen's inequality for Lagrangian submanifolds of ℂPn(4) as discovered by Opreaffi We show that, for n≥4, an n-dimensional Lagrangian submanifold in ℂPn(4) for which equality is attained at all points is necessarily minimal.

We prove the following two new optimal immersion results for complex projective space. First, if $n\equiv3\,\Mod 8$ but $n\not\equiv3\,\Mod 64$, and $\alpha(n)=7$, then $CP^{n}$ can be immersed in $\mathbb{R}^{4n-14}$. Second, if $n$ is even and $\alpha(n)=3$, then $CP^n$ can be immersed in $\mathbb{R}^{4n-4}$. Here $\alpha(n)$ denotes the number of 1s in the binary expansion of $n$. The first contradicts a result of Crabb, which said that such an immersion does not exist, apparently due to an arithmetical mistake. We combine Crabb's method with that developed by the author and Mahowald.

We give a formula for the Laplacian of the second fundamental form of an n-dimensional compact minimal submanifold M in a complex projective space CPm. As an application of this formula, we prove that M is a geodesic minimal hypersphere in CPm if the sectional curvature satisfies K≥1/n, if the normal connection is flat, and if M satisfies an additional condition which is automatically satisfied when M is a CR submanifold. We also prove that M is the complex projective space CPn/2 if K≥3/n, and if the normal connection of M is semi-flat.

In this paper, we characterize hypersurfaces of type A2 in a complex projective space in terms of their geodesics.