If you want to move the geometry service from the Utilities folder of your ArcGIS Server site to another folder, you'll need to delete the service and re-create it in the desired folder using the ArcGIS Server Administrator Directory.
To re-create the geometry service, follow these steps:
- In a web browser, open the Administrator Directory. The URL is formatted http://gisserver.domain.com:6080/arcgis/admin and log in.
- In the Administrator Directory, click services.
- In the folders list, select the folder that will contain the geometry service. If you don't see an appropriate folder, you can create one by choosing createFolder.
- Click createService.
- On the Operation - createService page, paste the following JSON (JavaScript Object Notation) into the Service Properties text box:
- Click Create.Your geometry service is created and appears in the Services list. 
- Optionally, start the geometry service by clicking Geometry and clicking start on the Service - Geometry (GeometryServer) page.
{
 "serviceName": "Geometry",
 "type": "GeometryServer",
 "description": "The Geometry service is used by application developers to assist with geometric operations such as projecting, generalizing, and intersecting geometry.",
 "capabilities": "null",
 "minInstancesPerNode": 1,
 "maxInstancesPerNode": 2,
 "instancesPerContainer": 1,
 "maxWaitTime": 60,
 "maxStartupTime": 300,
 "maxIdleTime": 1800,
 "maxUsageTime": 600,
 "loadBalancing": "ROUND_ROBIN",
 "isolationLevel": "HIGH",
 "configuredState": "STOPPED",
 "recycleInterval": 24,
 "recycleStartTime": "00:00",
 "keepAliveInterval": -1,
 "private": false,
 "isDefault": false,
 "maxUploadFileSize": -1,
 "allowedUploadFileTypes": "",
 "properties": {},
 "extensions": [],
 "datasets": []
}Caution:
Do not change the value for serviceName. Clients will be unable to consume your geometry service if it is named something other than Geometry.