Question:
As we know for a two-dimensional surface the eigenvalues of its shape operator are the minimum and maximum of the corresponding quadratic form (second fundamental form); they indicate the maximum and minimum of the normal curvature.
Now I want to know whether there exists an analogous geometrical interpretation for the eigenvalues of shape operator of an n-dimensional hypersurface or not.
Definitions:
The first fundamental form $I$ of a surface element is just the restriction of the Euclidean inner product in $\mathbb R^n$ to all tangent hyperplanes $T_uf$, i.e., $$I(X,Y):=\langle X,Y \rangle$$ for any two tangent vectors $X,Y \in T_uf$ or for vectors $X,Y \in \mathbb R^n$ which are tangent to the surface element$.^1$
Shape operator of a surface is the minus derivative of the unit normal vectors on the surface. Formally speaking, let $f:U \to \mathbb {R}^3$ be a surface element with unit normal vector map $\nu$, $\nu: U \to S^2$ is defined by $$\nu (u_1,u_2):=\frac{\frac{\partial f}{\partial u_1} \times \frac{\partial f}{\partial u_2}}{\left \Vert \frac{\partial f}{\partial u_1} \times \frac{\partial f}{\partial u_2} \right \Vert},$$
then for every $u\in U$ we have the linear map $$D\nu|_u:T_uU \to T_uf,$$ where $T_uU=\{u\} \times \mathbb R^2$ and $T_uf=Df|_u\left(T_uU\right)$, and $$Df|_u:T_uU \to T_uf$$ is a linear isomorphism. Then the shape operator $L:=-D\nu \circ (Df)^{-1}$ is defined pointwise by $$L_u:=-\left(D\nu|_u \right) \circ \left(Df|_u\right)^{-1}:T_uf \to T_uf\,.^2$$ The above definition can be easily generalized to the general $\mathbb R^n$ space.
Let $f:U \to \mathbb R^3$ be given. Then for tangent vectors $X$ and $Y$, one defines:
the second fundamental form $I\!I$ of $f$ by $$I\!I(X,Y):=I(LX,Y),$$ where $L$ is the shape operator$.^3$
The above definition can be easily generalized to the general $\mathbb R^n$ space.
[1], [2], [3] Wolfgang Kühnel, "Differential Geometry Curves-Surfaces-Manifolds", Second Edition, American Mathematical Society, 2006.
The second fundamental form is essentially the best quadratic approximation to the surface at that point (after applying a rigid motion so that the critical point is the origin and the tangent plane is horizontal). Up to an orthogonal change of basis, any quadratic form can be written in diagonal form, as $$ Q(u) = \lambda_1 (u_1)^2 + \dots + \lambda_n(u_n)^2. $$ So the eigenvalues provide a geometric description of this quadratic form, and thus an approximate geometric description of the shape of the surface near that point.