I will be teaching some "topology" to high school students. I was wondering how to explain to such a school student that on a sphere the shortest path between 2 points is given by a great circle?
Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot find a nice way to convince them since they see the sphere embedded on 3d?
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If you connect the two points by a rubber band in the shape of a meandering path on the sphere, it is intuitive the rubber band will snap into a great circle shape.
Alternatively you can explain the geodesic as the path a magnetic marble would take if the sphere were a steel ball, and you let the marble roll along the surface of the sphere. It will roll in a great circle path, without veering "left" or "right."
I'm not sure what you mean about not seeing above or below, since the sphere is orientable?
I'm still not sure what you mean by "looking on the sky or ground." You mean that we are used to thinking of the sphere as having codimension 1, which is meaningless if we think of the sphere as an abstract manifold rather than embedded in space?
I suppose you could argue by analogy to the circle. If you draw a circle on paper, you can travel perpendicular from the circle in two directions, inside and outside. But this is an accident of the fact that the circle is drawn on the paper. If instead you form a hoop in 3D, there is no longer two ways of moving away from the circle -- there are many directions you can travel, and the hoop no longer separates space into any kind of inside or outside.
I suppose to really blow their minds you could show the horned sphere...
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You can always rotate the sphere so that points A and B are both on the equator. The idea then is you reduce your distance from point B the fastest if you head in the direction of point B, and that direction is along the equator.
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There are two ways I tried with students.
Case 1. Equator of ball
On a plastic ball toy carefully tie a string around any great circle, ( use a smal cellulose tape/tab if needed, to prevent side slippage ,) for exactly one rotation. Make the string taut by pulling in opposite directions. The ball will be compressed, tension in taut string increases.
Case 2. Parallel circles of ball
Next repeat the same by marking a smaller or latitude circle, stick the string on it. When pulled the string easily slips out of its place.
Case 3. Winding and unwinding a cone
Next roll a rectangular sheet of paper into a cone. Look at the edges in two situations when rolled and when flattened out. The sideways straightness is preserved in either case... as zero curvature.
In Case 1 a sidewise straightness existed, in Case 2 no sidewise straightness existed, in fact the edge became base of a cone.
These concepts of geodesy and geodesic curvature in differential geometry can be thus demonsrated.
Next, up-down feeling is conditioned by 1) Gravity or Force vector 2) Fluid sensation in Cochlea semi-circular coils inside human ear as a biological response to such forces..
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Use a ball. Note that the less curvy a line is, the straighter it is (and therefore shorter).
Then note that cutting a slice through the middle of the ball gets you the straightest line available. Which is a good definition of a great circle.
(And as other answerers have pointed out, if the ball is edible then you will be nourishing bodies as well as minds).
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Set up lines of latitude and longitude on the sphere so that the two points, call them $A$ and $B$, are on the same meridian.
Then any motion from $A$ to $B$ can be "tracked" along the meridian by taking a moving point on the meridian that has the same latitude as the point moving from $A$ to $B$. The motion along the meridian (an arc of a great circle) is shorter, because at every instant it sweeps out length at a rate less than or equal to that of any other motion at the same latitudes.
In the extreme case, when the points are antipodal, this is all still true but the inequality becomes an equality if the other motion is along a different great circle.
Also, how to explain that if they lived on a sphere they would have no notion of "above" or "below"? I cannot find a nice way to convince them since they see the sphere embedded on 3d?
They do live on a sort of sphere. Do two people on opposite sides of the Earth have a way of deciding which one is upside down? Assigning directions North/South or East/West relies on extra information such as a magnetic pole or the motion of the sun relative to the Earth.
I have found it helpful to replace the sphere by an apple and introduce an "internal" observer by placing an ant on the apple. The ant will crawl from point $A$ to point $B$ on the sphere by following the shortest path (the queen can't wait) which is always an arc of great circle.
An additional point that students find illuminating is the phenomenon that a plane on a direct flight from New York to Paris will veer rather far North instead of following the same latitude throughout the flight. This is of course also because the latitude is not a minimizing path (except for the equator).