We know that we can have a noncompact surface with a so called Monkey saddle (<https://en.wikipedia.org/wiki/Monkey_saddle>), on which there is a isolated flat umbilic (where the Gaussian curvature vanishes), and the Gaussian curvature is strictly negative everywhere else.


So my question is, is there an example where there is a monkey type saddle on a COMPACT Riemannian surface with nonpositive curvature? That is, can there be an isolated flat umbilic on it? Is it possible to somehow "compactify" the above noncompact exmaple without adding positive curvature part to it?



Isolate flat umbilic on compact Riemannian surface with nonpositive curvature
Compact surfaces with non-negative Euler characteristic cannot carry such metrics because of Gauss-Bonnet. However, this is the only obstruction.


First, such metrics exist on any compact orientable surface of genus greater than 1. For example, let $M$ be a hyperelliptic Riemann surface of genus $g\ge 2$, and let $\omega\_1,\ldots,\omega\_g$ be a basis for the holomorphic $1$-forms on the surface. Then the Riemannian metric
$$
g = \omega\_1\circ\overline{\omega\_1} + \cdots + \omega\_g\circ\overline{\omega\_g}
$$ 
has Gaussian curvature $K$ that is negative everywhere except at the Weierstrass points (of which there are $2g{+}2$), where it vanishes.


Second, one can construct such metrics on non-orientable surfaces with negative Euler characteristic: Consider a hyperelliptic Riemann surface $S$ of genus $g>1$ that has a fixed-point-free, anti-holomorphic involution $\iota$ with the property that there exists a basis $\omega\_1,\ldots,\omega\_g$ as above for which $\iota^\*\omega\_k = \overline{\omega\_k}$. (Such examples exist for each $g>1$.) Then the above metric $g$ is invariant under $\iota$ and hence descends to be a metric with the desired properties on the non-orientable surface that one obtains by quotienting $S$ by the $\mathbb{Z}\_2$-action generated by $\iota$. Since this construction can be done for each $g>1$, this gives a sequence of non-orientable compact examples covering all negative Euler characteristics.


