A submanifold $M^m$ of a generalized Sasakian-space-form ${\overline{M}}^{2n+1}(f_1,f_2,f_3)$ is said to be $C$-totally real submanifold if $\xi \in \Gamma \left(T^{\perp }M\right)$ and $\phi X\in \Gamma \left(T^{\perp }M\right)$ for all $X\in \Gamma \left(TM\right)$. In particular, if $m=n$, then $M^n$ is called Legendrian submanifold. Here, we derive Wintgen inequalities on Legendrian submanifolds of generalized Sasakian-space-forms with respect to different connections; namely, quarter symmetric metric connection, Schouten-van Kampen connection and Tanaka-Webster connection.

The aim of this paper is to determine explicitly the algebraic structure of the curvature algebra of the 3-dimensional Heisenberg group with left invariant cubic metric. We show, that this curvature algebra is an infinite dimensional graded Lie subalgebra of the generalized Witt algebra of homogeneous vector fields generated by three elements.

We prove that Witten’s Conjecture [40] on the relationship between the Donaldson and Seiberg-Witten series for a four-manifold of Seiberg-Witten simple type with $b_1=0$ and odd $b_2^{+}\ge 3$ follows from our $\left(3\right)$-monopole cobordism formula [6] when the four-manifold has $c_1^2\ge {\chi }_h-3$ or is abundant.