It is not currently possible to use feature layers hosted in ArcGIS Online with curve geometry. During the publishing process, curves are converted to Rings because curves are currently not supported geometry types for feature services hosted in ArcGIS Online.
For example, after publishing a polygon with curves from ArcGIS Desktop to ArcGIS Online, the array of curves are stored as Rings rather than curveRings. Instead of a curve, a large number of coordinates are created in the Rings to make the polygon appear curved.
More specifically, if a circular feature is created in ArcMap and the feature is converted to JSON, when viewing the JSON created by the ArcMap geoprocessing tool, curveRings with just a few coordinates are seen:
However, after publishing the same feature as a hosted feature service and querying the geometry from REST, the geometry shows as Rings with thousands of coordinates:
Ring0:[-8168422.4016999993,24143556.7408],[-8162128.594278127,24143555.093085036],[-8155834.7885817392,24143550.14994058], ... 11998 more...
Although curves are not included in the current release of ArcGIS Online, the ability to edit curves on hosted feature layers is planned to be implemented in a future release. To view more information about future ArcGIS Online updates, refer to the following help page: What's new: ArcGIS Online.
Alternatively, true curves are supported as geometry objects in a feature service hosted in ArcGIS for Server 10.3 or later. The query parameter returns true curves in output geometries when the returnTrueCurves parameter is set to true. This is because the JSON curve object is only supported for services hosted in ArcGIS for Server 10.3 or later, as documented on the ArcGIS REST API: Query (Map Service\Layer) help page. The following resource discusses the JSON curve object further: Geometry objects.