  We study the local differential geometry of varieties $X^n\subset \Bbb C\Bbb
P^{n+a}$ with degenerate secant and tangential varieties. We show that the
second fundamental form of a smooth variety with degenerate tangential variety
is subject to certain rank restrictions. The rank restrictions imply a slightly
refined version of Zak's theorem on linear normality and a short proof of the
Zak-Fantecchi theorem on the superadditivity of multisecant defects. We show
there is a vector bundle defined over general points of $TX$ whose fibers carry
the structure of a Clifford algebra. This structure implies additional
restrictions of the size of the secant defect. The Clifford algebra structure,
combined with further local computations, yields a new proof of Zak's theorem
on Severi varieties that is substantially shorter than the original. We also
prove local and global results on the dimension of the Gauss image of
degenerate tangential varieties, refining the results in [GH].
