The open-source mobile operating system built on the Linux kernel is Android.
Android was developed initially by Android Inc. and is now primarily developed and maintained by Google, with the Open Handset Alliance involved in its early platform development. Its first commercial release, Android 1.0, arrived in 2008. The system was designed chiefly for touchscreen devices such as smartphones and tablets, and later expanded to televisions, cars, watches, and other embedded products.
Android uses a modified Linux kernel beneath its application framework, libraries, hardware interfaces, and Android Runtime. That makes Android Linux-based, but it is not identical to a typical desktop Linux distribution: Android uses its own userland components, including the Bionic C library and Toybox utilities, rather than the usual GNU desktop stack.
The phrase “open-source” needs a little context. The Android Open Source Project publishes much of the platform’s source code, but many commercial devices also include proprietary components such as Google Mobile Services. Symbian, Windows Phone, and iOS are different mobile operating systems and are not Android alternatives built on this same Linux foundation.