Address: Sør-Trøndelag University College,
E-mail: h.j.rivertz@gmail.com
Abstract: We will prove that if an open subset of $\mathbb{C}{}P^{n}$ is isometrically immersed into $\mathbb{C}{}P^{m}$, with $m<(4/3)n-2/3$, then the image is totally geodesic. We will also prove that if an open subset of $\mathbb{H}{}P^{n}$ isometrically immersed into $\mathbb{H}{}P^{m}$, with $m<(4/3)n-5/6$, then the image is totally geodesic.
AMSclassification: primary 53C40.
Keywords: submanifolds, homogeneous spaces, symmetric spaces.