Sectional curvatures under simple maps Suppose that we have a submanifold $X$ of $\mathbb{R}^n$ with the induced Euclidean metric, whose sectional curvatures we have a handle of (say, they are lower bounded by some $\kappa$). 
Is there a way to get a handle on the sectional curvatures of simple transformations of $X$ (say, lower bound them again)? Maybe the easiest transformation would be a linear map (so, the curvatures of $A X$, for some, say invertible, matrix $A \in \mathbb{R}^{n \times n}$). 
I've tried to do this in a brute-force manner, through the second fundamental form, and the definition of sectional curvature in terms of the second fundamental form, but it doesn't seem to yield anything useful.
I also find the question interesting for other types of curvature, e.g. Ricci. 
 A: Given the map $u: X \rightarrow \mathbb{R}^n$ and coordinates $x^1, \dots, x^k$ on $X$, the second fundamental form is a normal-vector-valued symmetric $2$-tensor on $X$. In other words, for each $x \in X$, it is a tensor $H(x) \in T^\perp_xX\otimes S^2T_xX$. The simplest formula I know for it is the following:
$H(x)$ is the normal component of $\partial^2u(x)$. Therefore, if $H = H_{ij}dx^idx^j$, then
$$ H_{ij} = \partial^2_{ij}u - g^{pq}(\partial_pu)(\partial_qu\cdot\partial^2_{ij}u), $$
where
$$ g_{ij} = \partial_iu\cdot \partial_j u $$
are the components of the induced Riemannian metric on $X$ and
$$ \Gamma^k_{ij} - g^{kp}\partial_pu\cdot\partial^2_{ij}u $$
are the Christoffel symbols.
Given a linear transformation $A: \mathbb{R}^n \rightarrow \mathbb{R}^n$, you can set $\tilde{u} = Au$. It's now straightforward to compute the second fundamental form $\widetilde{H}$ of $\tilde{u}$.
This also can be used for computing the second fundamental form for more general transformations of the submanifold, but I don't know how enlightening such formulas would be.
