In the first paper of this series we studied on a compact regular contact manifold the integral of the Ricci curvature in the direction of the characteristic vector field considered as a functional on the set of all associated metrics. We showed that the critical points of this functional are the metrics for which the characteristic vector field generates a 1-parameter group of isometries and conjectured that the result might be true without the regularity of the contact structure. In the present paper we show that this conjecture is false by studying this problem on the tangent sphere bundle of a Riemannian manifold. In particular the standard associated metric is a critical point if and only if the base manifold is of constant curvature +1 or −1; in the latter case the characteristic vector field does not generate a 1-parameter group of isometries.

Simply connected conformally flat Riemannian manifolds are characterized as hypersurfaces in the light cone of a standard flat Lorentzian space, transversal to its generators. Some applications of this fact are given.

We study real hypersurfaces of a complex projection space and show that there are no such hypersurfaces with harmonic curvature on which the structure vector is principal.

We consider the extent of certain complete hypersurfaces of Euclidean space. We prove that every complete hypersurface in En+1 with sectional curvature bounded below and non-positive scalar curvature has at least (n − 1) unbounded coordinate functions.

Let M be a smooth surface in Euclidean space E3 and L the Weingarten map. The fundamental forms I1, I2, I3,… on M are defined in terms of L and the usual inner product 〈, 〉 of E3 as follows. If X and Y are in the tangent space TPM of M (Pε M), then I1(X, Y) = 〈X, Y), I2(X, Y) = 〈LX, Y〉, I3(X, Y) = 〈L2X, Y), etc. Moreover, if M is convex, i.e., the Gaussian curvature K = k1k2, where ki, (i = l,2) are the principal curvatures of M, is everywhere positive, then one can also define on M the forms I0(X, Y) = 〈L−1X, Y), I−1,(X, Y) = 〈L−2X, Y), I−2(X, Y) = 〈 L−3X, Y) etc., where L−1 is the inverse of L. Since L is self-adjoint, the forms Im are, for any integer m, symmetric bilinear functions on TPM × TPM. Furthermore Im are C∞ in the sense that if X and Y are vector fields with domain A ⊂ M, then 〈 LmX, Y〉P = 〉LmXP, YP) is a C∞ real function on A. If the convex surface M is appropriately oriented, then the forms Im define metrics on M, which we also denote by 〈, 〉m (〈, 〉1)≡ 〈, 〉).

The study of the integral of the scalar curvature, ∫MRdVg, as a function on the set of all Riemannian metrics of the same total volume on a compact manifold is now classical, and the critical points are the Einstein metrics. On a compact contact manifold we consider this and ∫M (R − R* − 4n2) dv, with R* the *-scalar curvature, as functions on the set of metrics associated to the contact structure. For these integrals the critical point conditions then become certain commutativity conditions on the Ricci operator and the fundamental collineation of the contact metric structure. In particular, Sasakian metrics, when they exist, are maxima for the second function.

In this paper we consider how much we can say about an irreducible symmetric space M which admits a single hypersurface with at most two distinct principal curvatures. Then we prove that if N is conformally flat, then N is quasiumbilical and M must be a sphere, a real projective space or the noncompact dual of a sphere or a real projective space.

The problem is the reconstruction of the shape of an object, whose shell is a surface star-shaped with respect to a point 0, from the knowledge of the volume of every “half-object” obtained by taking any plane through 0. Conditions for the existence and uniqueness of the solution are given. The main result consists in showing that any uniform a-priori bound on the mean curvature of the shell reestablishes continuous dependence on the data for bodies satisfying a certain symmetry condition.

Defining a function on the set of all Riemannian metrics associated to a contact form on a compact manifold by taking the integral of the Ricci curvature in the direction of the characteristic vector field, it is shown that on a compact regular contact manifold the only critical points of this function are the metrics for which the characteristic vector field generates a group of isometrics.

In 1903 H. Minkowski obtained two integral formulae for closed convex surfaces in three dimensional Euclidean space. In this paper we obtain generalised Minkowski formulae on compact orientable immersed submanifolds of an arbitrary Riemannian manifold. By successive specialisation we indicate how known integral theorems can be obtained as particular cases of our result.

A sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

Let a compact orientable manifold be immersed as a hypersurface of constant mean curvature in an Einstein space. It is shown that the immersion is totally umbilical if and only if there exists a conformal variation of the immersion whose variation vector is nowhere tangential to the hypersurface.

Let N be a complete connected Riemannian manifold with sectional curvatures bounded from below. Let M be a complete simply connected Riemannian manifold with sectional curvatures KM(σ)≤ −a2 (a ≥ 0) and with dimension < 2 dim N. Suppose that N is isometrically immersed in M and that its image lies in a closed ball of radius ρ. Then sup(KN(σ) − KM(σ)) ≥ μ2(aρ)/ρ2 where the function μ is defined by μ(x) = x coth x for x > 0, μ(0) = 1 and the supremum is taken over all sections tangent to N.

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if its first and second fundamental forms are proportional. In this paper we prove the following best possible result.

Let Mn be a smooth, compact and strictly convex, embedded hypersurface of Rn + 1 (n ≥ 1), an ovaloid for short. By “strictly convex” we mean that the Gauss-Kronecker curvature where ki are the principal curvatures with respect to the inner unit normal field, is everywhere positive. It is well knpwn [5, p. 41] that, for such a hypersurface, the spherical-image mapping is a diffeomorphism onto the unit hypersphere. Furthermore, Mn is the boundary of an open bounded convex body, which we shall call the interior of Mn.

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

Totally umbilical submanifolds of dimension greater than four in quaternion-space-forms are completely classified.

Consider two convex bodies K, K′ in Euclidean space En and paint subsets β, β′ on the boundaries of K and K′. Now assume that K′ undergoes random motion in such a way that it touches K.

The even dimensional C∞ manifold M2n is said to be symplectic, if there exists a 2-form Ω, defined everywhere on M such that

(i)Ω is closed, that is dΩ, = 0, and

(ii)Ωn ≠ 0.