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Orthogonal Partial Conformal Change


If we start with a generic nonnegatively curved metric on a Riemannian manifold and apply a deformation to the metric, then nonnegative curvature is usually not preserved. From the perspective of systems of equations, the problem of finding a metric with a prescribed curvature tensor is a highly overdetermined system. However, if every zero curvature plane is tangent to a totally geodesic flat that is fixed by a deformation, then the problem of finding the prescribed metric becomes more manageable. In 2008 Petersen and Wilhelm exploited this idea and used it to develop a deformation called Orthogonal Partial Conformal Change (OPCC). Then they successfully combined it with other deformations to put a positively curved metric on the Gromoll-Meyer sphere.

Here we introduce new examples of manifolds that admit an OPCC deformation. In particular, we generalize a result obtained by Walschap on the canonical line bundle over the complex projective space. One peculiar characteristic of one of our OPCC deformations is that they are not verifiable via the machinery developed by Petersen and Wilhelm.

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