|Question: What is the influence of Buddhism on Bhagavad Gita?|
The hInayAna Buddhism (thEravAda) does not seem to have much influence on Gita.
Epic Hinduism developed from Veda Hinduism. Bhagavad Gita is a part of the Epic Mahabharata and hence a part of Epic Hinduism. The three epics (Mahabharata, Mahabhagavata (Purana?), and Ramayana) did not totally disapprove sacrifices with vEdic chants.
There was no place for devotion (bhakti) in vEdic sacrifices. Epics not only approved devotion but also canvassed for and encouraged it.
mAhAyAna Buddhism also encouraged "devotion" - the devotion towards Buddha.
mAhAyAna buddhism considered Buddha as an incarnation of God. Mahabharata canvassed Krishna as the incarnation of God. Ramayana canvassed Rama as the incarnation of God. Mahabhagavata detailed the ten incarnations of Lord vishNu.
Hinduism started accepting Buddha as the ninth incarnation of Lord vishNu.
Bhagavadgita did not speak of, or encourage idol worship. All the three epics MahAbhAgavata, MahAbhAgavata and rAmAyaNa are silent about idol worship. The idol worship seems to have started in the post-Harsha (Harsha was a 7th Century Emperor of India during whose rule Huen Tsang visited India.) period.
Hindus imitating the mAhAyAna Buddhist practice of idol worship, Buddhism seems to have lost its competitive edge in India.