Geometric Morse theory ( and its complex analogy) In the  literature are  there some  concept of  geometric  version of  Morse or  Picard  Lefschets theory? That is the  comparison of  level sets  as Riemannian  submanifold not  merely as  topological manifolds.
In  particular is  there a  complete classification of Polynomials  $P(z,w): \mathbb{C}^2 \to  \mathbb{C}$  such that all regular  level sets are  mutually isometric  Riemannian manifolds  when  we consider them as $2$  dimensional  submanifolds  of  $\mathbb{R}^4 \simeq  \mathbb{C}^2$  where level  sets  inherit the  standard metric  $dx_1^2+dy_1^2+dx_2^2+dy_2^2$  of  $\mathbb{R}^4$? In particular  does  $z^2+w^2$ satisfy this property?  Here we identify $(z,w)\in \mathbb{C}^2$ with $(x_1,y_1,x_2,y_2) \in \mathbb{R}^4$ with $z=x_1+iy_1,\;\;w=x_2+iy_2$.
 A: Here is an answer for the last question: The regular level set $L_c := \{(z,w) \in \mathbb{C}^2 : P(z,w) = c \}$ of $P(z,w)=z^2 + w^2$ is not isometric to the level set $L_{2c}$. To see this, notice that $L_c$ and $L_{2c}$ are complete Riemannian manifolds. The map $(z,w) \to (\sqrt{2}z,\sqrt{2}w)$ restricts to a homothety $\phi:L_c \to L_{2c}$ which is not an isometry.  If $h: L_{2c} \to L_c$ is an isometry then $h \circ \phi$ is a homothetic transformation of $L_c$ which is not an isometry. Hence $L_c$ is locally Euclidean (see Lemma 2, page 242, Kobayashi-Nomizu Vol. I, 1963). Notice that the level sets $L_c$ are minimal submanifolds of $\mathbb{C}^2$. Finally, it is well-known (at least to the experts) that a minimal and Ricci flat submanifold of $\mathbb{R}^n$ is totally geodesic. Thus if $L_c$ and $L_{2c}$ are isometric then $z^2 + w^2 - c  = 0$ is a complex line. This is a contradiction since the affine complex curve $L_c$ is regular and has degree 2 whilst complex lines had degree 1. Then $L_c$ and $L_{2c}$ are not isometric as Riemannian manifolds.   
