If I lie on my stomach on a mirror-free frozen lake, how many cm is the earth curved at a distance of 1 kilometer?

Answer

So we say that the Earth is a perfectly smooth sphere.
Look at the attached figure. The circle is the earth, you are yourself in point P, and you are looking towards S.

The radius of the earth : MP = 6 371 km.

The distance PS is 1 km,

Now just with Pythagorean in the triangle MPS :

distance MS = square root [  (6 371 )² + (1)² ] = 6371 . 000 078 km

The red piece is therefore 0.000 078 km long, or 78 mm
At a distance of 1 km, the earth’s surface has therefore sunk 78 mm