Osculating spaces of intersection of two varieties Let $Z = X\cap Y\subset\mathbb{C}^N$ be a manifold given as the intersection of two manifolds $X,Y$ intersecting transversally along $Z$. Let $T_p^kX,T_p^kY,T_p^kZ$ be the $k$-osculating spaces at $p\in Z$ of $X,Y,Z$ respectively.
Is it true that $T^k_pZ = T^k_pX\cap T^k_pY$? I know the answer is positive for $k=1$ i.e. for tangent spaces.   
 A: No, it does not hold for higher order osculating spaces.
In the projective space $\mathbb{P}^{5}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^3$, and let $C\subset\mathbb{P}^3$ be a degree $3$ rational normal curve. Fixed an isomorphism $\psi:\mathbb{P}^1\rightarrow C$ we consider the rational normal scroll
$$S_{(1,3)} = \bigcup_{p\:\in\: \mathbb{P}^1}\left\langle p, \psi(p)\right\rangle\subset \mathbb{P}^{5}$$
where $\left\langle p, \psi(p)\right\rangle$ is the line through $p$ and $\psi(p)$. Then $S_{(1,3)}$ can be locally parametrized by the map
$$
\begin{array}{cccc}
\phi: & \mathbb{A}^1\times\mathbb{P}^1 & \longrightarrow & \mathbb{P}^{5}\\ 
 & (u,[\alpha_0:\alpha_1]) & \mapsto & [\alpha_0 u:\alpha_0:\alpha_1 u^3:\alpha_1 u^{2}:\alpha_1 u:\alpha_1].
\end{array} 
$$ 
Now, consider the Segre embedding 
$$
\begin{array}{cccc}
\sigma: & \mathbb{P}^1\times\mathbb{P}^3 & \longrightarrow & \mathbb{P}^{7}\\ 
 & ([u:v],[\alpha_0:\dots:\alpha_3]) & \mapsto & [\alpha_0u:\dots:\alpha_3u:\alpha_0v:\dots :\alpha_3v].
\end{array} 
$$ 
and let $\Sigma_{(1,3)}$ be its image. Note that $\Sigma_{(1,3)}$ is locally parametrized by
$$
\begin{array}{cccc}
\widetilde{\sigma}: & \mathbb{A}^1\times\mathbb{P}^3 & \longrightarrow & \mathbb{P}^{7}\\ 
 & ([u:1],[\alpha_0:\dots:\alpha_3]) & \mapsto & [\alpha_0u:\dots:\alpha_3u:\alpha_0:\dots :\alpha_3].
\end{array} 
$$ 
and that $\deg(\Sigma_{(1,3)}) = \deg(S_{(1,3)}) = 4$. Now, take $\alpha_{i} = \alpha_1u^{i-1}$ for $i=2,3$. Then
$$\widetilde{\sigma}(u,[\alpha_0:\alpha_1:\dots :\alpha_1u^{2}]) = [\alpha_0u:\alpha_1u:\alpha_1u^{2}:\alpha_1u^3:\alpha_0:\alpha_1:\dots:\alpha_1u^{2}]$$
and the coordinates functions of this last map are exactly the ones appearing in the expression of $\phi$. Therefore, if $[Z_0:\dots:Z_{7}]$ are the homogeneous coordinates on $\mathbb{P}^{7}$ and 
$$H^{5} = \{Z_j-Z_{5+j}=0,\: j = 1,2\}\cong\mathbb{P}^{5}$$
then we have 
$$S_{(1,3)} = \Sigma_{(1,3)}\cap H^{5}\subset\mathbb{P}^{7}$$
If $p\in S_{(1,3)}$ is a general point then $\dim(T_p^2S_{(1,3)}) = 4$. On the other hand, $T_p^2\Sigma_{(1,3)} = \mathbb{P}^{7}$. We conclude that
$$T_p^2S_{(1,3)}\subsetneqq T_p^2\Sigma_{(1,3)} \cap H^{5} = H^{5}$$
