A Generalisation of Obata’s theorem
نویسنده
چکیده
In a complete Riemannian manifold (M,g) if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of (M,g). In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds (M,g) admitting a real valued function u such that the hessian of u has atmost two eigenvalues −u and − 2 , under some mild hypothesis on (M,g). This generalises a well known result of Obata which characterizes all round spheres.
منابع مشابه
An Obata-type Theorem on a Three-dimensional Cr Manifold
We prove a CR version of the Obata’s result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three dimensional manifold with non-negative CR-Paneitz operator which satisfies a Lichnerowicz type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of t...
متن کاملLinearisation of w-Proofs: ANew Method of Generalisation within Automated Deduction
Generalisation is a major 'open' problem in theorem-proving which must often be addressed when attempting automation of proofs involving mathematical induction. This paper proposes a new, uniform method of generalisation, involving the transformation of proofs, which encompasses many different types of generalisation and which may succeed when other methods fail.
متن کاملAn Obata Type Result for the First Eigenvalue of the Sub-laplacian on a Cr Manifold with a Divergence-free Torsion
We prove a CR version of the Obata’s result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian manifold which satisfies a Lichnerowicz type condition and has a divergence free pseudohermitian torsion. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the pse...
متن کاملSingular and Totally Singular Generalised Quadratic Forms
In this paper we present a decomposition theorem for generalised quadratic forms over a division algebra with involution in characteristic 2. This is a generalisation of a decomposition result on quadratic forms in characteristic 2 from [3] and extends a generalisation of the Witt decomposition theorem for nonsingular forms to cover forms that may be singular.
متن کاملLimit theorems for projections of random walk on a hypersphere
We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein-Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a “functional” generalisa...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008