Morse height function for general compact manifold

Question

Can you give me the form of the height function for any compact manifold embedded in the reals?

Maybe the projection of the parametrization onto a basis vector ex.

For the n-sphere is $h(p)=\pi(\phi(p))=x_{n+1}$.

thanks

Answer 1

If $\phi:M \to \Bbb R^n$ is in fact an isometric embedding, then the image $\text{Im} \phi$ of $M$ under $\phi$ is compact, since $M$ is compact. If the $x_i$, $1 \le i \le n$, are a set of Cartesian coordinates on $\Bbb R^n$ (where by "Cartesian" I mean compatible with the vector space structure on $\Bbb R^n$, so they are related by a nonsingular linear transformation to the coordinates of the standard basis), the the $n$ functions $x_i \circ \phi$, being continuous (even differentiable!) functions on the compact set $M$, each attain a minimum $m_i$ and maximum $M_i$ in $M$. If we consider the hyperplane $x_i = m_i$ in $\Bbb R^n$, then for $p \in M$, $x_i \circ \phi(p) - m_i$ is the height of the point $\phi(p)$ above this hyperplane. It is correct to assert the height functions associated with a given Cartisian coordinate system in $\Bbb R^n$ are given by the projections onto the coordinate axes, offset by the positions or "heights", relative to $0$, of the hyperplanes themselves. In fact, one needn't restrict oneself to the the offsets being the $m_i$, though the hyperplanes $x_i = m_i$ have the nice feature that $\phi(M)$ "sits" right on them! Taking any hyperplane $x_i = a_i$ yields a height function $x_i \circ \phi - a_i$ relative to it. One can of course even generalize to the case of an arbitrary hyperplane $H$ given by an equation of the form $\mathbf n \cdot \mathbf r = a$, where $\mathbf n$ is the unit normal vector to $H$ and $a$ is the distance 'twixt $H$ and $0 \in \Bbb R^n$; then $\mathbf n \cdot \phi(p) - a$ is the height of $\phi(p)$ relative to $H$; since $p \in M$, $\mathbf n \cdot \phi - a$ is a function on $M$ itself.

It should be observed however that such height functions won't necessarily be Morse functions in the sense of having non-singular Hessians at their critical points, etc. That depends on the specifics of $M$ and $\phi$.

Hope this helps. Cheers,

and of course,

Fiat Lux!!!