• Akhil Ranjan

      Articles written in Proceedings – Mathematical Sciences

    • On invariant convex cones in simple Lie algebras

      S Kumaresan Akhil Ranjan

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      This paper is devoted to a study and classification ofG-invariant convex cones ing, whereG is a lie group andg its Lie algebra which is simple. It is proved that any such cone is characterized by its intersection withh-a fixed compact Cartan subalgebra which exists by the very virtue of existence of properG-invariant cones. In fact the pair (g,k) is necessarily Hermitian symmetric.

    • Substantial Riemannian submersions ofS15 with 7-dimensional fibres

      Akhil Ranjan

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      In this paper we show that a substantial Riemannian submersion ofS15 with 7-dimensional fibres is congruent to the standard Hopf fibration. As a consequence we prove a slightly weak form of the diameter rigidity theorem for the Cayley plane which is considerably stronger than the very recent radius rigidity theorem of Wilhelm.

    • An intrinsic approach to Lichnerowicz conjecture

      Akhil Ranjan

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      In this paper we give a proof of Lichnerowicz conjecture for compact simply connected manifolds which is intrinsic in the sense that it avoids thenice embeddings into eigenspaces of the Laplacian. Even if one wants to use these embeddings, this paper gives a more streamlined proof. As a byproduct, we get a simple criterion for a polynomial to be a Jacobi polynomial.

    • Convexity of spheres in a manifold without conjugate points

      Akhil Ranjan Hemangi Shah

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      For a non-compact, complete and simply connected manifoldM without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres inM is a radial function, then the geodesic spheres are convex. We also show that ifM is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index lemma.

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