Without modifying core or creating a custom Manage Profile module you cannot achieve this. 

However, if you are tempted to modify core and risk having to do this again in the future ... the file to edit is located under 

admin\Users\User.ascx.vb and you can achieve this by adding just 3 lines:

Case "firstname", "lastname", "displayname", "email"
                    If IsUser Then e.Editor.Visible = False

Add this to this this part of the file:

''' -----------------------------------------------------------------------------
        Protected Sub UserEditorCreated(ByVal sender As Object, ByVal e As UI.WebControls.PropertyEditorItemEventArgs) Handles UserEditor.ItemCreated
            Dim setting As Object = Nothing

            Select Case e.Editor.Name.ToLower
                Case "firstname", "lastname", "displayname", "email"
                    If IsUser Then e.Editor.Visible = False
                Case "displayname"
                    setting = UserModuleBase.GetSetting(UserPortalID, "Security_DisplayNameformat")
                    If (Not setting Is Nothing) AndAlso (Not String.IsNullOrEmpty(Convert.ToString(setting))) Then
                        If AddUser Then
                            e.Editor.Visible = False
                        Else
                            e.Editor.EditMode = PropertyEditorMode.View
                        End If
                    End If
                Case "email"
                    setting = UserModuleBase.GetSetting(UserPortalID, "Security_EmailValidation")
                    If (Not setting Is Nothing) AndAlso (Not String.IsNullOrEmpty(Convert.ToString(setting))) Then
                        e.Editor.ValidationExpression = Convert.ToString(setting)
                    End If
            End Select
        End Sub

Good luck

Can anyone tell me how I can do the same thing for a field that is located in the Manage Profile section?  I'd like to make that one invisible as well! 

Thanks