Totally η-umbilic real hypersurfaces are the simplest examples of real hypersurfaces in a non-flat complex space form. Geodesic hyperspheres in this ambient space are typical examples of such real hypersurfaces. We characterise every geodesic hypersphere by observing the extrinsic shapes of its geodesics and using the derivative of its contact form.

The goal of this paper is to describe explicitly simple modules for Schur superalgebra S(2|2) over an algebraically closed field K of characteristic zero or positive characteristic p>2.

A group G is n-central if Gn ≤ Z(G), that is the subgroup of G generated by n-powers of G lies in the centre of G. We investigate pk-central groups for p a prime number. For G a finite group of exponent pk, the covering group of G is pk-central. Using this we show that the exponent of the Schur multiplier of G is bounded by $p^{\lceil \frac{c}{p-1} \rceil}$, where c is the nilpotency class of G. Next we give an explicit bound for the order of a finite pk-central p-group of coclass r. Lastly, we establish that for G, a finite p-central p-group, and N, a proper non-maximal normal subgroup of G, the Tate cohomology Hn(G/N, Z(N)) is non-trivial for all n. This final statement answers a question of Schmid concerning groups with non-trivial Tate cohomology.

Let COP = 0 ∩ H∞, where 0 is the little Bloch space on the open unit disk , and A() be the disk algebra on . For non-zero functions u1,u2,. . ., uN ∈ A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕN satisfying ϕj ∈ A() and ∥ϕj∥∞=1 for every j, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕj maps COP into A().

Let R be a commutative Noetherian ring, is an ideal of R and M is an R-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that =0 for all i>magRM. We prove this in several cases.

We give the necessary and sufficient conditions for Lagrangian submanifolds in Kähler manifolds to be biharmonic. We classify biharmonic PNMC Lagrangian H-umbilical submanifolds in the complex space forms. Furthermore, we classify biharmonic PNMC Lagrangian surfaces in the two-dimensional complex space forms.