On the manifoldM(M) of all Riemannian metrics on a compact manifold M, one can consider the natural L2-metric as described first by [11]. In this paper we consider variants of this metric, which in general are of higher order. We derive the geodesic equations; we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.

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In this work, we investigate wave transmission property through a zero index metamaterial (ZIM) waveguide embedded with rectangular dielectric defects. We show that total reflection and total transmission (cloaking) can be achieved by adjusting the geometric sizes and/or permittivities of the defects. Our work provides another possibility of manipulating wave propagation through ZIM in addition to the widely studied dielectric defects with cylindrical geometries.

In this paper, we consider the L p dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the L p dual Minkowski problem when pq ≥ 0 and the given data is even. If f ≡ 1, we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.