Well, not every IP address has a name. However, given the IPAddress
you can use Dns.GetHostEntry
to try to resolve it. Also note that if it's being a NAT router, you'll be getting the router's IP address rather than their actual machine.
And just to address the point in the comments, I agree that there's no point in ToString/Parse/ToString:
IPAddress address = ((IPEndPoint)_client.Client.RemoteEndPoint).Address;
Globals.connectedIPAddress = address.ToString();
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