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. 



Osculating spaces of intersection of two varieties
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
$$T\_p^2S\_{(1,3)}\subsetneqq T\_p^2\Sigma\_{(1,3)} \cap H^{5} = H^{5}$$


