Note - I'm excluding a centrifugal-force type explanation for this question, as explained below, even though I provide a possible explanation of how that may function .

THE QUESTION
Why do  geostationary-satellites  fall to earth at the N or S pole, and yet not at the equator ?

THE QUESTION IN MORE DETAIL
How would a satellite / object possibly physically detect the difference between being positioned above the N or S pole,  or,  above the equator, since I'm logically ruling out a centrifugal-force type explanation.

LOGICALLY RULING OUT CENTRIFUGAL-FORCE
   You would logically think the laws relating to CENTRIFUGAL-FORCE would only apply to objects physically touching the earth, there is no cable attached between the satellite and the earth .

A POSSIBLE  EXPLANATION WHICH COULD ACTUALLY MAKE A CENTRIFUGAL-FORCE EXPLANATION POSSIBLE
Maybe it has something to do with -
   - The surface texture of the earth is uneven, consisting of things like mountains, valleys, etc, etc
   - ( Also, the shape of the earth is not perfectly spherical, it is like a very slightly squashed-ball due to centrifugal-force of it's own rotation, although that may not be relevant to this explanation . )
   -  So, maybe the uneven surface texture of the earth gives it's gravity some tread like a tyre, so that it enables the gravity to grip the satellite from more than one point, and the result is the centrifugal-force effect at the equator,  and not at the N or S pole since the earth does not rotate there . 

ANOTHER EXPLANATION
 - Is it possible that a geostationary-satellite above the equator does not fall to earth,  BECAUSE,  there are other objects in the universe with a gravitational-field ( sun, jupiter, saturn, moon ), providing other gravitational-anchors / gravitational-points-of-reference, even though these gravitational-anchors are not stationary .
     In other words, if the only two objects in the universe would be the geostationary-satellite above the equator, and,  the earth,  would the satellite still not fall to earth above the equator.
      - This could only be tested in  gravity-simulation-software, not anywhere else, since gravitational-anchors are everywhere .

ANOTHER EXPLANATION  -  INTERACTION BETWEEN GRAVITY AND AETHER
 - Is it possible that a geostationary-satellite above the equator does not fall to earth,  BECAUSE, maybe there is an interaction between  gravity and 'aether', SO that because the earth rotates at the equator, this interaction effect between  gravity and 'aether' is magnified / multiplied,  preventing a geostationary-satellite above the equator from falling to earth. ( strangely, a repelling force, anti-gravity effect !!!!!!!!!!!!!!!!!!!  )
 - Of course, the earth is also travelling through space ( around the sun ), but I don't know how to effectively work that into the theory I'm typing 'in this post' .

   WHEN THE MOON PASSES OVER THE N OR S POLE, DOES IT TEMPORARILY FALL TO EARTH  ?
   - Have any unusual effects ever been detected ?

   THE FOLLOWING IS THEORETICAL, IF THE EARTH'S-AXIS-OF-ROTATION POINTED AT THE SUN .
   WHEN THE MOON PASSES OVER THE N OR S POLE,  AND,  THE EARTH IS DIRECTLY BETWEEN THE MOON AND THE SUN ( OR SATURN OR JUPITER INSTEAD OF THE SUN ), IS THERE AN INCREASED EFFECT OF THE MOON TEMPORARILY FALLING TO EARTH  ?
    -  YES, I assume obviously the N or S pole never faces the sun, HOWEVER -
    -  What about jupiter or saturn
    -  Have any unusual effects ever been detected ?

   Are there any unusual-effects / experiments that can be observed right on the N or S pole, WHICH DO NOT INVOLVE GYROSCOPES, instead, just something like a rolling-ball or something, or maybe some type of aircraft  ?

   When any particular type of aircraft flies to the N or S pole, from any point outside of the N or S pole, are there any unusual-effects that may have been detected, APART FROM INCREASED GRAVITY  ?

  Do any particular type of aircraft anywhere on earth experience any unusual effects, apart from increased gravity approaching the N or S pole,  effects which are unexplained, relating to this post ?

   Did proffessor Laithwaite fully test BOTH the VERTICALLY and HORIZONTALLY positioned gyroscope for anti-gravity effect ?
   - I may be wrong, I thought he did actually detect a decrease in weight in some tests.

   WILL A GYROSCOPE SPIN FOR AN  'IDENTICAL'  AMOUNT OF TIME, WHEN IT IS POSITIONED VERTICALLY,  COMPARED TO WHEN IT IS POSITIONED HORIZONTALLY  ?
   - Of course, to do this this test properly, the gyroscope should be on a  100%-FULLY-LEVITATED bearing, and maybe also the whole device should be sitting on a  100%-FULLY-LEVITATED  floor,  maybe even on a floating-aircraft,  due to small earth vibrations .

  DISTANCE TRAVELLED OF A BALL PROPELLED INTO THE AIR INSIDE A MOVING TRAIN,  COMPARED TO A BALL PROPELLED FROM STATIONARY POSITION
   - Would an experiment measuring distance travelled of a ball propelled into the air COMPLETELY INSIDE / COMPLETELY ENCLOSED inside a moving train,  compared to a ball propelled into the air from a stationary position on earth, give predicted/calculated results ( using the laws ), or, unusual results ?   
    ( Of course taking into account that the earth is both rotating-on-it's-own-axis and precession etc )
 
   IF YOU ARE EXACTLY ON THE EQUATOR, AND YOU PROPEL A BALL INTO THE AIR IN AN EXACT WEST DIRECTION, AND THEN COMPARE THE DISTANCE TRAVELED TO THE BALL PROPELLED IN AN EXACT EAST DIRECTION, WILL THE TWO DISTANCES TRAVELLED BE THE SAME ?

  WHAT IS THE EFFECT THAT EXPLAINS WHY THE  'AXIS-OF-THE-EARTH'S-ROTATION-ON-IT'S-OWN-AXIS'  IS AT A SPECIFIC ALMOST 90-DEGREE-ANGLE( OR 180-DEGREE-ANGLE ) IN RELATION TO THE SUN ?
  - Is this effect replicated by the moon in relation to the earth, or by the other planets, etc.

  Is it possible satellites 'secretly' use the solar-panels for propulsion or steering, by running current through them at some frequency ( being careful not to emit any visible light ) ?

   TIME / FRAMES-OF-REFERENCE   ? -  WHEN APPLYING THE LAWS OF MOMENTUM OR GRAVITY TO LARGE THINGS, LIKE AN OBJECT IN A FAR AWAY GALAXY TRAVELLING MUCH FASTER THAN THE SPEED OF LIGHT COMPARED TO THE POSITION OF THE EARTH
    - Although I don't know what I am talking about here, and have not researched it, don't the laws of momentum or gravity calculate things in very very small time intervals ( nanoseconds ?, or much much smaller ?,  or no time ? ),  SO, what about comparing the effectiveness / validity of these laws in two different scenarios -
        -  A ball being swung on a 1-meter string at 200km/h
        -  An object in a far away galaxy travelling at 100-light-years-per-second when compared to the position of the earth ( although you could instead say the earth is travelling at 100-light-years-per-second )
   In one time-interval / frame-of-reference ( nanoseconds ?, or much much smaller ?,  or no time ? ),  the ball has not moved at all, and yet the object in a far away galaxy has traveled thousands of kilometers ( assuming a frame-of-reference has a time component ).   Maybe the time component is variable with ( in proportion to ) some distance component .

   COULD AN INTERACTION BETWEEN THE MAGNETIC ( OR ELECTRIC OR GRAVITATIONAL ) FIELD OF EARTH,   'WITH',   AETHER IN SPACE,  THROUGH WHICH THE EARTH TRAVELS AND ROTATES,   BE GENERATING POWER ( ELECTRICITY OR OTHER ) ONTO THE EARTH,  LIKE,  THE ROTATING-PERMANENT-MAGNET-DISK IN THE  'FARADAY-PARADOX',  COULD IT MAYBE EXPLAIN LIGHTNING / AURORAS
    - Obviously I have not researched this, I'm only posting it, to post it.
    - ( Yes, I have read about particle friction in clouds, and energy from the sun etc, but other theories are always useful )
    - What about the moon, is there unexplained electrical activity on it, that could be explained by this theory, or could it explain electrical activity on other planets.
    - If this theory does generate power ( electricity or other power ) onto the earth, does this power somehow leave earth, or does the earth( and moon etc ) charge up like a capacitor.