# Article

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Keywords:
submanifolds; homogeneous spaces; symmetric spaces
Summary:
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.
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