example of compact manifold with $K\le 0$

Question

Could you give an example of such Riemannian manifold: 1. compact; 2. non-positive sectional curvature

Answer 1

The usual example is a "flat torus": $[0,1]^2$ with the top and bottom, as well as left and right, edges identified.

There are similar examples with constant negative curvature, but they are a bit more involved to describe.

One possibility is to take two regular hyperbolic right-angled octagons and glue them together along opposite edges, producing a cylinder-like thing with two longitudinal seams where each end has four right-angled corners. You can then glue each of these edges shut by identifying parts of them in the flat-torus pattern.

      B--3--C              Q--3--R
     /       \            /       \
    4         4          5         5
   /           \        /           \
  A             D      P             S
  |             |      |             |
  1             2      2             1
  |             |      |             |
  H             E      W             T
   \           /        \           /
    7         7          8         8
     \       /            \       /
      G--6--F              V--6--U

The resulting manifold is a genus-2 surface. It has four points where four of the original octagon corners come together:

    |           |           |           |
    5           2           1           8
   Q|R         P|D         T|H         U|V
-3--+--3-   -5--+--4-   -8--+--7-   -6--+--6-
   C|B         S|A         W|E         G|F
    4           1           2           7
    |           |           |           |

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A simpler construction would be to take a larger hyperbolic octagon with corner angle 45°, and simply identify opposite sides (preserving orientation). This also produces a compact genus-2 surface with constant negative curvature.

Answer 2

Take the unit disc $D$ in $\Bbb C$ with hyperbolic metric, then factor it by a discrete group of Mobius transformations acting freely on $D$. If this group has the property that the quotient is compact, then you have an example. (All compact Riemann surfaces of genus $\ge2$ can be obtained this way.)