Tangential Connection

Let $X\subset\P^N$ be a smooth variety. The embedding in $\P^n$ gives naturally rise to the notion of embedded tangent spaces. That is the locus spanned by tangent lines to a point $x\in X$. Generally the embedded tangent space intersects the variety $X$ only at the point $x$. In this paper I am interested in those $X$ for which this intersection, for $x\in X$ general, is a positive dimensional subvariety. The results of this paper support the conjecture that these varieties are built out of some special varieties that I call {\sl Tangentially Connected}, see Definition \ref{def:TC}. Actually I prove this under mild restrictions.