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I have a polygon feature layer and a polygon geometry. How do I find out if the feature layer contains the polygon, as in exact shape of the polygon?

I did see the geometry engine, but that doesn't contain a method for knowing if exact shape exists at same place. Both datasets have same spatial reference.

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You could create a QueryTask on the layer, and specify the geometry of the polygon in the Query's geometry clause and a SPATIAL_REL_INTERSECTS for the query type.

This would return any geometries from the layer which intersect with the polygon.

You could then examine those to see whether they had matching start/end coordinates, number of vertices, etc. Or convert each polygon to JSON and compare with the original polygon.

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