• K Ramachandran

      Articles written in Proceedings – Mathematical Sciences

    • Harmonic manifolds with some specific volume densities

      K Ramachandran A Ranjan

      More Details Abstract Fulltext PDF

      We show that a noncompact, complete, simply connected harmonic manifold (Md, g) with volume densityθm(r)=sinhd-1r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M2d, g) with volume densityθm(r)=sinh2d-1r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.

    • Finite dimensional imbeddings of harmonic spaces

      K Ramachandran A Ranjan

      More Details Abstract Fulltext PDF

      For a noncompact harmonic manifoldM we establish finite dimensionality of the eigensubspacesVγ generated by radial eigenfunctions of the form coshr+c. As a consequence, for such harmonic manifolds, we give an isometric imbedding ofM into (Vγ,B), whereB is a nondegenerate symmetric bilinear indefinite form onVγ (analogous to the imbedding of the real hyperbolic spaceHn into ℝn+1 with the indefinite formQ(x,x)=−x02+Σxi2). This imbedding is minimal in a ‘sphere’ in (Vγ,B). Finally we give certain conditions under whichM is symmetric.

  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2021-2022 Indian Academy of Sciences, Bengaluru.