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To understand the local conditions, it's convenient to establish canonically associated local coordinate expressions for the quantities involved. Thus, let $(M^2,g,X)$ and $(N^2,h,Y)$ be as described and suppose that we want to test whether, for a given $p\in M$ and $q\in N$, there exists an open $p$-neighborhood $U\subset M$ and a local diffeomorphism $f:U\to N$ satisfying $f(p) = q$ with the desired properties. To simplify notation a little bit, let us fix orientations on $M$ and $N$ and require $f$ to be orientation-preserving. (We'll see what comes of this choice later.)

Then it is easy to show that there exist oriented, $p$-centered coordinates $(r,\theta):V\to (-\epsilon, \epsilon)\times (-\epsilon,\epsilon)$ on an open $p$-neighborhood $V\subset M$ and a smooth function $A:(-\epsilon,\epsilon)\to\mathbb{R}$ with $A(0)=0$ and $A'>0$ such that, on $V$, we have $g = \mathrm{d}r^2 + A'(r)^2\,\mathrm{d}\theta^2$ and $X = \partial/\partial\theta$. Similarly, there exist oriented, $q$-centered coordinates $(s,\phi):V\to (-\delta, \delta)\times (-\delta,\delta)$ on an open $q$-neighborhood $W\subset M$ and a smooth function $B:(-\delta,\delta)\to\mathbb{R}$ with $B(0)=0$ and $B'>0$ such that, on $W$, we have $h = \mathrm{d}s^2 + B'(s)^2\,\mathrm{d}\phi^2$ and $Y = \partial/\partial\phi$. Such adapted coordinates are locally unique.

[Note that, since $M$ is simply-connected (and, let's assume, connected, though the OP didn't include that condition), the functions $r$ and $\theta$ extend globally to $M$ uniquely so that $g = \mathrm{d}r^2 + a^2\,\mathrm{d}\theta^2$ and $X=\partial/\partial\theta$ where $a$ is a positive smooth function on $M$ satisfying $\mathrm{d}r\wedge\mathrm{d}a = 0$. The smooth mapping $(r,\theta):M\to\mathbb{R}^2$ is an immersion, but there is no reason to believe, under the given hypotheses, that it is an embedding, nor is it necessarily true that $a = A'(r)$ for some function $A:r(M)\to\mathbb{R}$. The situation with $(N,h,Y)$ is similar.]

Supposing that an $f$ exists with all the specified properties, we can, by shrinking $\epsilon$, assume that $f(V)\subset W$ and hence, using the fact that $f_\ast(X) = Y$, conclude that $$ f^*(s) = s\circ f = R(r)\quad\text{and}\quad f^*(\mathrm{d}\phi) = \mathrm{d}\theta + M(r)\,\mathrm{d}r.\tag1 $$ for some functions $R$ and $M$ on $(-\epsilon,\epsilon)$ with $R(0)=0$ and $R'>0$. This implies that, relative to the orthonormal coframings, we must have $$ f^*\begin{pmatrix}\mathrm{d}s\\ B'(s)\,\mathrm{d}\phi\end{pmatrix} = \begin{pmatrix}R'(r) & 0\\ B'(R(r))M(r) & B'(R(r))/A'(r)\end{pmatrix} \begin{pmatrix}\mathrm{d}r\\ A'(r)\,\mathrm{d}\theta\end{pmatrix}\tag2 $$

Now, the constancy of the singular values implies that, in particular, the determinant of the above coefficient matrix must be constant, i.e., that there must be a constant $c_2>0$ such that $$ R'(r)B'(R(r))/A'(r) = c_2\,.\tag3 $$ Since $B(0) = R(0) = A(0) = 0$, we then integrate to get $B(R(r)) = c_2\,A(r)$. In particular, since $B$ is invertible, $R(r) = B^{-1}\bigl(c_2\,A(r)\bigr)$ for some positive constant $c_2$.

Now, the sum of the squares of the singular values of the coefficient matrix must be another constant $c_1 > 2c_2$ (so that the two constant singular values will be distinct) such that $$ R'(r)^2 + B'(R(r))^2\,M(r)^2 + B'(R(r))^2/A'(r)^2 = c_1\,.\tag4 $$ Using the above formula for $R'(0) = c_2 A'(0)/B'(0)$, we see that, by taking $c_1$ sufficiently large, we can guarantee that the above equation for $M(r)$ has (two) real solutions on a neighborhood of $r=0$.

Conversely, for $c_2>0$ and $c_1 > 2c_2$ sufficiently large, there will be functions $R(r)$ and $M(r)$ that satisfy the above equations (3) and (4) and the initial condition $R(0)=0$, and hence, via (1) and the initial condition $f(p)=q$, they will determine a unique mapping $f$ with the desired properties.

Thus, local mappings $f$ with constant, distinct singular values always exist carrying any desired point to any other. Moreover, it is clear that there is a 2-parameter family of such local mappings carrying any given point in the domain to any given point in the range.

The existence of a global such mapping $f:M\to N$ depends on the growth properties of the functions $A$ and $B$ and the validity of their domains. Little more can be said about this without more information or hypotheses about the functions $A$ and $B$.

To understand the local conditions, it's convenient to establish canonically associated local coordinate expressions for the quantities involved. Thus, let $(M^2,g,X)$ and $(N^2,h,Y)$ be as described and suppose that we want to test whether, for a given $p\in M$ and $q\in N$, there exists an open $p$-neighborhood $U\subset M$ and a local diffeomorphism $f:U\to N$ satisfying $f(p) = q$ with the desired properties. To simplify notation a little bit, let us fix orientations on $M$ and $N$ and require $f$ to be orientation-preserving. (We'll see what comes of this choice later.)

Then it is easy to show that there exist oriented, $p$-centered coordinates $(r,\theta):V\to (-\epsilon, \epsilon)\times (-\epsilon,\epsilon)$ on an open $p$-neighborhood $V\subset M$ and a smooth function $A:(-\epsilon,\epsilon)\to\mathbb{R}$ with $A(0)=0$ and $A'>0$ such that, on $V$, we have $g = \mathrm{d}r^2 + A'(r)^2\,\mathrm{d}\theta^2$ and $X = \partial/\partial\theta$. Similarly, there exist oriented, $q$-centered coordinates $(s,\phi):W\to (-\delta, \delta)\times (-\delta,\delta)$ on an open $q$-neighborhood $W\subset M$ and a smooth function $B:(-\delta,\delta)\to\mathbb{R}$ with $B(0)=0$ and $B'>0$ such that, on $W$, we have $h = \mathrm{d}s^2 + B'(s)^2\,\mathrm{d}\phi^2$ and $Y = \partial/\partial\phi$. Such adapted coordinates are locally unique.

[Note that, since $M$ is simply-connected (and, let's assume, connected, though the OP didn't include that condition), the functions $r$ and $\theta$ extend globally to $M$ uniquely so that $g = \mathrm{d}r^2 + a^2\,\mathrm{d}\theta^2$ and $X=\partial/\partial\theta$ where $a$ is a positive smooth function on $M$ satisfying $\mathrm{d}r\wedge\mathrm{d}a = 0$. The smooth mapping $(r,\theta):M\to\mathbb{R}^2$ is an immersion, but there is no reason to believe, under the given hypotheses, that it is an embedding, nor is it necessarily true that $a = A'(r)$ for some function $A:r(M)\to\mathbb{R}$. The situation with $(N,h,Y)$ is similar.]

Supposing that an $f$ exists with all the specified properties, we can, by shrinking $\epsilon$, assume that $f(V)\subset W$ and hence, using the fact that $f_\ast(X) = Y$, conclude that $$ f^*(s) = s\circ f = R(r)\quad\text{and}\quad f^*(\mathrm{d}\phi) = \mathrm{d}\theta + M(r)\,\mathrm{d}r.\tag1 $$ for some functions $R$ and $M$ on $(-\epsilon,\epsilon)$ with $R(0)=0$ and $R'>0$. This implies that, relative to the orthonormal coframings, we must have $$ f^*\begin{pmatrix}\mathrm{d}s\\ B'(s)\,\mathrm{d}\phi\end{pmatrix} = \begin{pmatrix}R'(r) & 0\\ B'(R(r))M(r) & B'(R(r))/A'(r)\end{pmatrix} \begin{pmatrix}\mathrm{d}r\\ A'(r)\,\mathrm{d}\theta\end{pmatrix}\tag2 $$

Now, the constancy of the singular values implies that, in particular, the determinant of the above coefficient matrix must be constant, i.e., that there must be a constant $c_2>0$ such that $$ R'(r)B'(R(r))/A'(r) = c_2\,.\tag3 $$ Since $B(0) = R(0) = A(0) = 0$, we then integrate to get $B(R(r)) = c_2\,A(r)$. In particular, since $B$ is invertible, $R(r) = B^{-1}\bigl(c_2\,A(r)\bigr)$ for some positive constant $c_2$.

Now, the sum of the squares of the singular values of the coefficient matrix must be another constant $c_1 > 2c_2$ (so that the two constant singular values will be distinct) such that $$ R'(r)^2 + B'(R(r))^2\,M(r)^2 + B'(R(r))^2/A'(r)^2 = c_1\,.\tag4 $$ Using the above formula for $R'(0) = c_2 A'(0)/B'(0)$, we see that, by taking $c_1$ sufficiently large, we can guarantee that the above equation for $M(r)$ has (two) real solutions on a neighborhood of $r=0$.

Conversely, for $c_2>0$ and $c_1 > 2c_2$ sufficiently large and $\epsilon>0$ sufficiently small, there will be functions $R(r)$ and $M(r)$ that satisfy the above equations (3) and (4) and the initial condition $R(0)=0$, and hence, via (1) and the initial condition $f(p)=q$, they will determine a unique mapping $f$ with the desired properties.

Thus, local mappings $f$ with constant, distinct singular values always exist carrying any desired point to any other. Moreover, it is clear that there is a 2-parameter family of such local mappings carrying any given point in the domain to any given point in the range.

The existence of a global such mapping $f:M\to N$ depends on the growth properties of the functions $A$ and $B$ and the validity of their domains. Little more can be said about this without more information or hypotheses about the functions $A$ and $B$.

To understand the local conditions, it's convenient to establish canonically associated local coordinate expressions for the quantities involved. Thus, let $(M^2,g,X)$ and $(N^2,h,Y)$ be as described and suppose that we want to test whether, for a given $p\in M$ and $q\in N$, there exists an open $p$-neighborhood $U\subset M$ and a local diffeomorphism $f:U\to N$ satisfying $f(p) = q$ with the desired properties. To simplify notation a little bit, let us fix orientations on $M$ and $N$ and require $f$ to be orientation-preserving. (We'll see what comes of this choice later.)

Then it is easy to show that there exist oriented, $p$-centered coordinates $(r,\theta):V\to (-\epsilon, \epsilon)\times (-\epsilon,\epsilon)$ on an open $p$-neighborhood $V\subset M$ and a smooth function $A:(-\epsilon,\epsilon)\to\mathbb{R}$ with $A(0)=0$ and $A'>0$ such that, on $V$, we have $g = \mathrm{d}r^2 + A'(r)^2\,\mathrm{d}\theta^2$ and $X = \partial/\partial\theta$. Similarly, there exist oriented, $q$-centered coordinates $(s,\phi):V\to (-\delta, \delta)\times (-\delta,\delta)$ on an open $q$-neighborhood $W\subset M$ and a smooth function $B:(-\delta,\delta)\to\mathbb{R}$ with $B(0)=0$ and $B'>0$ such that, on $W$, we have $h = \mathrm{d}s^2 + B'(s)^2\,\mathrm{d}\phi^2$ and $Y = \partial/\partial\phi$. Such adapted coordinates are locally unique.

Supposing that an $f$ exists with all the specified properties, we can, by shrinking $\epsilon$, assume that $f(V)\subset W$ and hence, using the fact that $f_\ast(X) = Y$, conclude that $$ f^*(s) = s\circ f = R(r)\quad\text{and}\quad f^*(\mathrm{d}\phi) = \mathrm{d}\theta + M(r)\,\mathrm{d}r.\tag1 $$ for some functions $R$ and $M$ on $(-\epsilon,\epsilon)$ with $R(0)=0$ and $R'>0$. This implies that, relative to the orthonormal coframings, we must have $$ f^*\begin{pmatrix}\mathrm{d}s\\ B'(s)\,\mathrm{d}\phi\end{pmatrix} = \begin{pmatrix}R'(r) & 0\\ B'(R(r))M(r) & B'(R(r))/A'(r)\end{pmatrix} \begin{pmatrix}\mathrm{d}r\\ A'(r)\,\mathrm{d}\theta\end{pmatrix}\tag2 $$

Now, the constancy of the singular values implies that, in particular, the determinant of the above coefficient matrix must be constant, i.e., that there must be a constant $c_2>0$ such that $$ R'(r)B'(R(r))/A'(r) = c_2\,.\tag3 $$ Since $B(0) = R(0) = A(0)$, we then integrate to get $B(R(r)) = c_2\,A(r)$. In particular, since $B$ is invertible, $R(r) = B^{-1}\bigl(c_2\,A(r)\bigr)$ for some positive constant $c_2$.

Now, the sum of the squares of the singular values of the coefficient matrix must be another constant $c_1 > 2c_2$ (so that the two constant singular values will be distinct) such that $$ R'(r)^2 + B'(R(r))^2\,M(r)^2 + B'(R(r))^2/A'(r)^2 = c_1\,.\tag4 $$ Using the above formula for $R'(0) = c_2 A'(0)/B'(0)$, we see that, by taking $c_1$ sufficiently large, we can guarantee that the above equation for $M(r)$ has (two) real solutions on a neighborhood of $r=0$.

Thus, for $c_2>0$ and $c_1 > 2c_2$ sufficiently large, there will be functions $R(r)$ and $M(r)$ that satisfy the above equations (3) and (4) and the initial condition $R(0)=0$, and hence, via (1) and the initial condition $f(p)=q$, they will determine a unique mapping $f$ with the desired properties.

Thus, local solutions always exist carrying any desired point to any other. Moreover, it is clear that there is a 2-parameter family of local solutions carrying any given point in the domain to any given point in the range.

The existence of a global solution $f:M\to N$ depends on the growth properties of the functions $A$ and $B$ and the validity of their domains. Little more can be said about this without more information or hypotheses.

To understand the local conditions, it's convenient to establish canonically associated local coordinate expressions for the quantities involved. Thus, let $(M^2,g,X)$ and $(N^2,h,Y)$ be as described and suppose that we want to test whether, for a given $p\in M$ and $q\in N$, there exists an open $p$-neighborhood $U\subset M$ and a local diffeomorphism $f:U\to N$ satisfying $f(p) = q$ with the desired properties. To simplify notation a little bit, let us fix orientations on $M$ and $N$ and require $f$ to be orientation-preserving. (We'll see what comes of this choice later.)

Then it is easy to show that there exist oriented, $p$-centered coordinates $(r,\theta):V\to (-\epsilon, \epsilon)\times (-\epsilon,\epsilon)$ on an open $p$-neighborhood $V\subset M$ and a smooth function $A:(-\epsilon,\epsilon)\to\mathbb{R}$ with $A(0)=0$ and $A'>0$ such that, on $V$, we have $g = \mathrm{d}r^2 + A'(r)^2\,\mathrm{d}\theta^2$ and $X = \partial/\partial\theta$. Similarly, there exist oriented, $q$-centered coordinates $(s,\phi):V\to (-\delta, \delta)\times (-\delta,\delta)$ on an open $q$-neighborhood $W\subset M$ and a smooth function $B:(-\delta,\delta)\to\mathbb{R}$ with $B(0)=0$ and $B'>0$ such that, on $W$, we have $h = \mathrm{d}s^2 + B'(s)^2\,\mathrm{d}\phi^2$ and $Y = \partial/\partial\phi$. Such adapted coordinates are locally unique.

[Note that, since $M$ is simply-connected (and, let's assume, connected, though the OP didn't include that condition), the functions $r$ and $\theta$ extend globally to $M$ uniquely so that $g = \mathrm{d}r^2 + a^2\,\mathrm{d}\theta^2$ and $X=\partial/\partial\theta$ where $a$ is a positive smooth function on $M$ satisfying $\mathrm{d}r\wedge\mathrm{d}a = 0$. The smooth mapping $(r,\theta):M\to\mathbb{R}^2$ is an immersion, but there is no reason to believe, under the given hypotheses, that it is an embedding, nor is it necessarily true that $a = A'(r)$ for some function $A:r(M)\to\mathbb{R}$. The situation with $(N,h,Y)$ is similar.]

Supposing that an $f$ exists with all the specified properties, we can, by shrinking $\epsilon$, assume that $f(V)\subset W$ and hence, using the fact that $f_\ast(X) = Y$, conclude that $$ f^*(s) = s\circ f = R(r)\quad\text{and}\quad f^*(\mathrm{d}\phi) = \mathrm{d}\theta + M(r)\,\mathrm{d}r.\tag1 $$ for some functions $R$ and $M$ on $(-\epsilon,\epsilon)$ with $R(0)=0$ and $R'>0$. This implies that, relative to the orthonormal coframings, we must have $$ f^*\begin{pmatrix}\mathrm{d}s\\ B'(s)\,\mathrm{d}\phi\end{pmatrix} = \begin{pmatrix}R'(r) & 0\\ B'(R(r))M(r) & B'(R(r))/A'(r)\end{pmatrix} \begin{pmatrix}\mathrm{d}r\\ A'(r)\,\mathrm{d}\theta\end{pmatrix}\tag2 $$

Now, the constancy of the singular values implies that, in particular, the determinant of the above coefficient matrix must be constant, i.e., that there must be a constant $c_2>0$ such that $$ R'(r)B'(R(r))/A'(r) = c_2\,.\tag3 $$ Since $B(0) = R(0) = A(0) = 0$, we then integrate to get $B(R(r)) = c_2\,A(r)$. In particular, since $B$ is invertible, $R(r) = B^{-1}\bigl(c_2\,A(r)\bigr)$ for some positive constant $c_2$.

Now, the sum of the squares of the singular values of the coefficient matrix must be another constant $c_1 > 2c_2$ (so that the two constant singular values will be distinct) such that $$ R'(r)^2 + B'(R(r))^2\,M(r)^2 + B'(R(r))^2/A'(r)^2 = c_1\,.\tag4 $$ Using the above formula for $R'(0) = c_2 A'(0)/B'(0)$, we see that, by taking $c_1$ sufficiently large, we can guarantee that the above equation for $M(r)$ has (two) real solutions on a neighborhood of $r=0$.

Conversely, for $c_2>0$ and $c_1 > 2c_2$ sufficiently large, there will be functions $R(r)$ and $M(r)$ that satisfy the above equations (3) and (4) and the initial condition $R(0)=0$, and hence, via (1) and the initial condition $f(p)=q$, they will determine a unique mapping $f$ with the desired properties.

Thus, local mappings $f$ with constant, distinct singular values always exist carrying any desired point to any other. Moreover, it is clear that there is a 2-parameter family of such local mappings carrying any given point in the domain to any given point in the range.

The existence of a global such mapping $f:M\to N$ depends on the growth properties of the functions $A$ and $B$ and the validity of their domains. Little more can be said about this without more information or hypotheses about the functions $A$ and $B$.

To understand the local conditions, it's convenient to establish canonically associated local coordinate expressions for the quantities involved. Thus, let $(M^2,g,X)$ and $(N^2,h,Y)$ be as described and suppose that we want to test whether, for a given $p\in M$ and $q\in N$, there exists an open $p$-neighborhood $U\subset M$ and a local diffeomorphism $f:U\to N$ satisfying $f(p) = q$ with the desired properties. To simplify notation a little bit, let us fix orientations on $M$ and $N$ and require $f$ to be orientation-preserving. (We'll see what comes of this choice later.)

Then it is easy to show that there exist oriented, $p$-centered coordinates $(r,\theta):V\to (-\epsilon, \epsilon)\times (-\epsilon,\epsilon)$ on an open $p$-neighborhood $V\subset M$ and a smooth function $A:(-\epsilon,\epsilon)\to\mathbb{R}$ with $A(0)=0$ and $A'>0$ such that, on $V$, we have $g = \mathrm{d}r^2 + A'(r)^2\,\mathrm{d}\theta^2$ and $X = \partial/\partial\theta$. Similarly, there exist oriented, $q$-centered coordinates $(s,\phi):V\to (-\delta, \delta)\times (-\delta,\delta)$ on an open $q$-neighborhood $W\subset M$ and a smooth function $B:(-\delta,\delta)\to\mathbb{R}$ with $B(0)=0$ and $B'>0$ such that, on $W$, we have $h = \mathrm{d}s^2 + B'(s)^2\,\mathrm{d}\phi^2$ and $Y = \partial/\partial\phi$. Such adapted coordinates are locally unique.

Supposing that an $f$ exists with all the specified properties, we can, by shrinking $\epsilon$, assume that $f(V)\subset W$ and hence, using the fact that $f_\ast(X) = Y$, conclude that $$ f^*(s) = s\circ f = R(r)\quad\text{and}\quad f^*(\mathrm{d}\phi) = \mathrm{d}\theta + M(r)\,\mathrm{d}r.\tag1 $$ for some functions $R$ and $M$ on $(-\epsilon,\epsilon)$ with $R(0)=0$ and $R'>0$. This implies that, relative to the orthonormal coframings, we must have $$ f^*\begin{pmatrix}\mathrm{d}s\\ B'(s)\,\mathrm{d}\phi\end{pmatrix} = \begin{pmatrix}R'(r) & 0\\ B'(R(r))M(r) & B'(R(r))/A'(r)\end{pmatrix} \begin{pmatrix}\mathrm{d}r\\ A'(r)\,\mathrm{d}\theta\end{pmatrix}\tag2 $$

Now, the constancy of the singular values implies that, in particular, the determinant of the above coefficient matrix must be constant, i.e., that there must be a constant $c_2>0$ such that $$ R'(r)B'(R(r))/A'(r) = c_2\,.\tag3 $$ Since $B(0) = R(0) = A(0)$, we then integrate to get $B(R(r)) = c_2\,A(r)$. In particular, since $B$ is invertible, $R(r) = B^{-1}\bigl(c_2\,A(r)\bigr)$ for some positive constant $c_2$.

Now, the sum of the squares of the singular values of the coefficient matrix must be another constant $c_1 > 2c_2$ (so that the two constant singular values will be distinct) such that $$ R'(r)^2 + B'(R(r))^2\,M(r)^2 + B'(R(r))^2/A'(r)^2 = c_1\,.\tag4 $$ Using the above formula for $R'(0) = c_2 A'(0)/B'(0)$, we see that, by taking $c_1$ sufficiently large, we can guarantee that the above equation for $M(r)$ has (two) real solutions on a neighborhood of $r=0$.

Thus, for $c_2>0$ and $c_1 > 2c_2$ sufficiently large, there will be functions $R(r)$ and $M(r)$ that satisfy the above equations, and hence they will determine the desired mapping $f$.

Thus, local solutions always exist carrying any desired point to any other. Moreover, it is clear that there is a 2-parameter family of local solutions carrying any given point in the domain to any given point in the range.

The existence of a global solution $f:M\to N$ depends on the growth properties of the functions $A$ and $B$ and the validity of their domains. Little more can be said about this without more information or hypotheses.

To understand the local conditions, it's convenient to establish canonically associated local coordinate expressions for the quantities involved. Thus, let $(M^2,g,X)$ and $(N^2,h,Y)$ be as described and suppose that we want to test whether, for a given $p\in M$ and $q\in N$, there exists an open $p$-neighborhood $U\subset M$ and a local diffeomorphism $f:U\to N$ satisfying $f(p) = q$ with the desired properties. To simplify notation a little bit, let us fix orientations on $M$ and $N$ and require $f$ to be orientation-preserving. (We'll see what comes of this choice later.)

Then it is easy to show that there exist oriented, $p$-centered coordinates $(r,\theta):V\to (-\epsilon, \epsilon)\times (-\epsilon,\epsilon)$ on an open $p$-neighborhood $V\subset M$ and a smooth function $A:(-\epsilon,\epsilon)\to\mathbb{R}$ with $A(0)=0$ and $A'>0$ such that, on $V$, we have $g = \mathrm{d}r^2 + A'(r)^2\,\mathrm{d}\theta^2$ and $X = \partial/\partial\theta$. Similarly, there exist oriented, $q$-centered coordinates $(s,\phi):V\to (-\delta, \delta)\times (-\delta,\delta)$ on an open $q$-neighborhood $W\subset M$ and a smooth function $B:(-\delta,\delta)\to\mathbb{R}$ with $B(0)=0$ and $B'>0$ such that, on $W$, we have $h = \mathrm{d}s^2 + B'(s)^2\,\mathrm{d}\phi^2$ and $Y = \partial/\partial\phi$. Such adapted coordinates are locally unique.

Supposing that an $f$ exists with all the specified properties, we can, by shrinking $\epsilon$, assume that $f(V)\subset W$ and hence, using the fact that $f_\ast(X) = Y$, conclude that $$ f^*(s) = s\circ f = R(r)\quad\text{and}\quad f^*(\mathrm{d}\phi) = \mathrm{d}\theta + M(r)\,\mathrm{d}r.\tag1 $$ for some functions $R$ and $M$ on $(-\epsilon,\epsilon)$ with $R(0)=0$ and $R'>0$. This implies that, relative to the orthonormal coframings, we must have $$ f^*\begin{pmatrix}\mathrm{d}s\\ B'(s)\,\mathrm{d}\phi\end{pmatrix} = \begin{pmatrix}R'(r) & 0\\ B'(R(r))M(r) & B'(R(r))/A'(r)\end{pmatrix} \begin{pmatrix}\mathrm{d}r\\ A'(r)\,\mathrm{d}\theta\end{pmatrix}\tag2 $$

Now, the constancy of the singular values implies that, in particular, the determinant of the above coefficient matrix must be constant, i.e., that there must be a constant $c_2>0$ such that $$ R'(r)B'(R(r))/A'(r) = c_2\,.\tag3 $$ Since $B(0) = R(0) = A(0)$, we then integrate to get $B(R(r)) = c_2\,A(r)$. In particular, since $B$ is invertible, $R(r) = B^{-1}\bigl(c_2\,A(r)\bigr)$ for some positive constant $c_2$.

Now, the sum of the squares of the singular values of the coefficient matrix must be another constant $c_1 > 2c_2$ (so that the two constant singular values will be distinct) such that $$ R'(r)^2 + B'(R(r))^2\,M(r)^2 + B'(R(r))^2/A'(r)^2 = c_1\,.\tag4 $$ Using the above formula for $R'(0) = c_2 A'(0)/B'(0)$, we see that, by taking $c_1$ sufficiently large, we can guarantee that the above equation for $M(r)$ has (two) real solutions on a neighborhood of $r=0$.

Thus, for $c_2>0$ and $c_1 > 2c_2$ sufficiently large, there will be functions $R(r)$ and $M(r)$ that satisfy the above equations (3) and (4) and the initial condition $R(0)=0$, and hence, via (1) and the initial condition $f(p)=q$, they will determine a unique mapping $f$ with the desired properties.

Thus, local solutions always exist carrying any desired point to any other. Moreover, it is clear that there is a 2-parameter family of local solutions carrying any given point in the domain to any given point in the range.

The existence of a global solution $f:M\to N$ depends on the growth properties of the functions $A$ and $B$ and the validity of their domains. Little more can be said about this without more information or hypotheses.