No, the Network Analyst extension is not required to build and edit a geometric network. A geometric network can be created with an ArcInfo or ArcEditor license.
A geometric network is used for modeling flow and is commonly applied to utilities datasets. Water distribution, electrical lines, gas pipelines, telephone services, and water flow in a stream are all examples of resource flows that can be modeled and analyzed using a geometric network.
More information on the Network Analyst extension can be found in the ArcGIS Desktop Web help in the Related Information below.