# Local obstructions for maps with constant singular values

$$\newcommand{\M}{\mathcal{M}}$$ $$\newcommand{\N}{\mathcal{N}}$$

Let $$\M, \N$$ be smooth two-dimensional Riemannian manifolds.

Are there any local obstructions for the existence of a smooth map $$f:\M \to \N$$ with constant distinct singular values?

(The singular values of $$df$$ can be defined to be those of any matrix representation of $$df$$ w.r.t orthonormal bases.)

If we replace the requirement of "constant distinct singular values" with "constant equal singular values" we recover the notion of homothetic manifolds (so there are obstructions for local equivalence, e.g. the curvature tensor).

If we replace it with "non-constant equal singular values" we get conformal equivalence, which in dimension $$2$$ has no obstruction (conformal flatness).

There are no local obstructions for constant distinct singular values when $$M$$ and $$N$$ have dimension $$2$$. This is locally a determined symmetric hyperbolic system of two equations for two unknowns, so it's always locally solvable.

Note 1: Local character of the equations

Giving a general analysis in all dimensions in classical PDE terms is not so easy notationally, although, for example, computing the linearization of the equations in the flat case at the particular solution $$f(x^1,\ldots,x^n) = (\lambda_1 x^1,\ \ldots,\ \lambda_n x^n) \tag1$$ where $$0<\lambda_1<\lambda_2<\cdots<\lambda_n$$ and seeing that this linearization is the system of $$n$$ equations $$\frac{\partial u^1}{\partial x^1} = \frac{\partial u^2}{\partial x^2} = \cdots = \frac{\partial u^n}{\partial x^n} = 0 \tag2$$ for $$u = \bigl(u^1(x),\ldots, u^n(x)\bigr)$$ is not hard.

For general metrics in $$n$$-dimensions, linearizing at a particular solution can be a bit messy, but, given a solution $$f$$ in a neighborhood of a point $$p\in M$$, one can always introduce $$p$$-centered coordinates $$x^i$$ and $$f(p)$$-centered coordinates $$y^i$$ on a neighborhood of $$f(p)\in N$$ so that, in these coordinates, $$f$$ near $$p$$ is of the form $$(1)$$ and so that the constant singular values PDE system for maps $$u$$ that are $$C^1$$-close to $$f$$ can be solved in the form $$\frac{\partial u^i}{\partial x^i} = E^i\left(x,u,\left[\frac{\partial u^j}{\partial x^k}\right]_{j\not=k}\right)\tag3$$ where the functions $$E^i$$ are smooth and vanish to second order at the origin. Thus, the linearization at $$u=f$$ will be of the form $$(2)$$. The point is that the coefficients of the actual metrics on $$M$$ and $$N$$ do not materially affect the form of the linearization.

Now, the system $$(3)$$ can be put in Cauchy form, and a solution $$u$$ near $$p$$ is completely determined by its restriction to a non-characteristic hypersurface $$H^{n-1}\subset M$$ passing through $$p$$, i.e., one whose conormal $$\xi_1\,\mathrm{d}x^1 + \cdots + \xi_n\,\mathrm{d}x^n = 0$$ at $$p$$ has $$\xi_1\xi_2\cdots\xi_n\not=0$$. Thus, the solutions depend locally on $$n$$ functions of $$n{-}1$$ variables, in the sense of Cauchy-Kowalewskaya.

When $$n=2$$, the fact that the characteristics are real and distinct implies that, in fact, the Cauchy problem is locally solvable in the smooth category. This result may generalize to $$n>2$$, but I'm not sure about that without checking the literature. In the real-analytic category, though, local solvability of the non-characteristic initial value problem immediately follows from the theorem of Cauchy-Kowalewskaya.

Note 2: The initial value problem

Here is what the above analysis, together with a little geometry, says about the initial value problem: Fix Riemannian $$n$$-manifolds $$(M^n,g)$$ and $$(N^n,h)$$ and an increasing sequence of positive constants: $$0<\lambda_1 <\lambda_2<\cdots <\lambda_n$$. Now let $$H^{n-1}\subset M$$ be a simply-connected smooth embedded hypersurface in $$M$$ and suppose that one can find a smooth map $$\phi:H\to N$$ such that the (not necessarily constant) singular values of $$\phi':TH\to TN$$ are smooth functions $$\mu_i:H\to \mathbb{R}$$ that satisfy $$\lambda_i < \mu_i < \lambda_{i+1}$$ for $$1\le i < n$$. Then one can show, by linear algebra that, for each $$p\in H$$ there exists $$2^n$$ choices for a linear map $$\psi:T_pM\to T_{\phi(p)}N$$ such that $$\psi(v) = \phi'(v)$$ for $$v\in T_pH$$ and the singular values of $$\psi$$ are $$\lambda_1,\ldots,\lambda_n$$. As $$p\in H$$ varies, these $$2^n$$ choices form a smooth covering space of degree $$2^n$$ over $$H$$. Since $$H$$ is simply-connected, it follows that one can choose a smooth isomorphism of vector bundles $$\Phi: H^*(TM)\to \phi^*(TN)$$ such that $$\Phi(v) = \phi'(v)$$ for $$v\in TH\subset H^*(TM)$$ and, for each $$p\in H$$, $$\Phi:T_pM\to T_{\phi(p)}N$$ has singular values $$\lambda_1,\ldots,\lambda_n$$. (There are $$2^n$$ such choices for $$\Phi$$).

Then the Cauchy-Kowalewskaya theorem says that, in the analytic category, there exists an open neighborhood $$U$$ of $$H$$ in $$M$$ and an analytic map $$f:U\to N$$ such that $$f(p) = \phi(p)$$ and $$f'(p) = \Phi(p)$$ for $$p\in H$$ and the singular values of $$f$$ are constant and equal to $$\{\lambda_1,\ldots,\lambda_n\}$$.

When $$n=2$$, this existence theorem holds in the smooth category. It might hold in the smooth category for all $$n$$, I don't know about that, but probably someone such as Deane Yang or Dennis DeTurck could determine this.

In any case, even in the smooth category, the existence result holds formally in the sense that, for each integer $$k\ge 0$$, there exists a smooth map $$f_k:U\to N$$ such that $$f_k(p) = \phi(p)$$ and $$f_k'(p) = \Phi(p)$$ and the singular values $$\sigma_1,\ldots,\sigma_n$$ of $$f_k$$ (which are not necessarily constant) have all their derivatives up to order $$k$$ vanish at $$p\in H$$, and this condition uniquely specifies the derivatives of $$f_k$$ up to order $$k{+}1$$. In particular, there is no local pointwise obstruction to the existence of an extension $$f$$ of $$\phi$$ that has constant singular values $$\{\lambda_1,\ldots,\lambda_n\}$$, since one can find an approximate solution (in fact, $$2^n$$ of them) to any desired order.

• @AsafShachar: Yes, I can describe the analysis if you like. You are correct that, in the non-flat cases, one does not have the explicit integration that one has in the flat cases, but the symbol of the PDE system is the same, so it is hyperbolic, as I explained in that case. Feb 6, 2021 at 16:01
• Dear Robert, thank you for this very interesting update. Very beautiful observation regarding the linearization's form. Just to be sure: (1) Since the solution $f$ is Iocally invertible - we can always find coordinates such that $f(x_1,\dots,x_n)=(x_1,\dots,x_n)$, even without the 'correct' factors $\lambda_i$. So do I guess correctly that you intended for the chosen coordinates to be isometric at the points $p, f(p)$, so the metrics of $M,N$ w.r.t. these coordinates at $p, f(p)$ would be $\delta_{ij}$? Feb 8, 2021 at 9:32
• (2) Doesn't your strategy show that given a solution there is a large family of solutions close to it? So a-priori shouldn't it be possible that there were no solutions to begin with? (What am I missing? I think that my rather poor PDE background might be showing here). Thank you again for all your patience and careful explanations. Feb 8, 2021 at 9:32
• @AsafShachar: (1) Yes, I'm adapting the coordinates to the metrics as well as assuming that $f$ is a local diffeomorphism. For example, we could take the $x$-coordinates to be $p$-centered geodesic normal coordinates and the $y$-coordinates to be $f(p)$-centered geodesic normal coordinates, each rotated so that, at the centers, they give the eigenvectors of the singular value decomposition. (2) Yes, it does show that, but, more, the C-K theorem proves existence as well (in the analytic category) as well as existence of formal power series solutions in the smooth case (convergence is an issue). Feb 8, 2021 at 10:49
• @AsafShachar: I'll put in a remark about the non-characteristic initial value problem. That should help. Feb 8, 2021 at 10:50