Who of the following founded a new city on the south bank of a tributary to river Krishna and undertook to rule his new kingdom as the agent of a deity to whom all the land south of the river Krishna was supposed to belong?

The correct answer is Harihara I.

Harihara I (also known as Hakka) was the founder of the Vijayanagara Empire along with his brother Bukka Raya I in 1336 CE. According to historical records and local legends, he established a new capital city, Vijayanagara (also called Vidyanagara in honor of his guru Vidyaranya), on the south bank of the Tungabhadra River, which is a major tributary of the river Krishna.
To legitimize his rule over the southern territories, Harihara I dedicated his kingdom to the deity Virupaksha (a form of Lord Shiva). He undertook to rule the newly established kingdom not as an independent monarch, but as a representative or agent (known as a pratinidhi) of Lord Virupaksha, to whom all the land south of the river Krishna was spiritually and temporally declared to belong.