Is Jesus, God the Father's representation? Is He His Avatar?
The short answer is no.
No Christian denomination employs such a term!
First of all, what is an avatar?
An avatar, a concept in Hinduism that means "descent", is the material appearance or incarnation of a deity on earth. The relative verb to "alight, to make one's appearance" is sometimes used to refer to any guru or revered human being.
The word avatar does not appear in the Vedic literature; however, it appears in developed forms in post-Vedic literature, and as a noun particularly in the Puranic literature after the 6th century CE. Despite that, the concept of an avatar is compatible with the content of the Vedic literature like the Upanishads as it is symbolic imagery of the Saguna Brahman concept in the philosophy of Hinduism. The Rigveda describes Indra as endowed with a mysterious power of assuming any form at will. The Bhagavad Gita expounds the doctrine of Avatara but with terms other than avatar.
Theologically, the term is most often associated with the Hindu god Vishnu, though the idea has been applied to other deities. Varying lists of avatars of Vishnu appear in Hindu scriptures, including the ten Dashavatara of the Garuda Purana and the twenty-two avatars in the Bhagavata Purana, though the latter adds that the incarnations of Vishnu are innumerable. The avatars of Vishnu are important in Vaishnavism theology. In the goddess-based Shaktism tradition of Hinduism, avatars of the Devi in different appearances such as Tripura Sundari, Durga and Kali are commonly found. While avatars of other deities such as Ganesha and Shiva are also mentioned in medieval Hindu texts, this is minor and occasional. The incarnation doctrine is one of the important differences between Vaishnavism and Shaivism traditions of Hinduism.
Incarnation concepts that are in some aspects similar to avatar are also found in Buddhism, Christianity, and other religions.
The Hindu understanding of avatars is similar to the Christian heresy of Docetism.
In the history of Christianity, docetism is the heterodox doctrine that the phenomenon of Jesus, his historical and bodily existence, and above all the human form of Jesus, was mere semblance without any true reality. Broadly it is taken as the belief that Jesus only seemed to be human, and that his human form was an illusion.
The word Δοκηταί ("Illusionists") referring to early groups who denied Jesus's humanity, first occurred in a letter by Bishop Serapion of Antioch (197–203), who discovered the doctrine in the Gospel of Peter, during a pastoral visit to a Christian community using it in Rhosus, and later condemned it as a forgery. It appears to have arisen over theological contentions concerning the meaning, figurative or literal, of a sentence from the Gospel of John: "the Word was made Flesh".
Docetism was unequivocally rejected at the First Council of Nicaea in 325 and is regarded as heretical by the Catholic Church, Eastern Orthodox Church, Coptic Orthodox Church of Alexandria, Orthodox Tewahedo, and many Protestant denominations that accept and hold to the statements of these early church councils, such as Reformed Baptists, Reformed Christians, all Trinitarian Christians.
The belief in Hindu system of avatars is very similar to the Christian heresy of Docetism, which is the belief that Jesus Christ only appeared to be human.
Thus Jesus could not be God the Father's representation are God the Father and God the Son with God the Holy Spirit are equally one God in three person and thus form the Holy Trinity. ThUs Jesus is not the Father’s Avatar!
That Jesus Christ is a God-man is generally acknowledged. That he is the Incarnation of God is dogma in the Christian religion; that he is an Avatar is to view him through Hindu eyes. In the West there has emerged a gradual recognition of Jesus Christ as an Avatar, and it developed circuitously, partly through secret societies, mystery schools, esoteric groups, and non-Christian religions such as Islam. It has already been shown how in the 1870s, Ramakrishna embraced Jesus as an Avatar and placed him within the Hindu pantheon.
"What is an avatar in Hinduism? Was Jesus an avatar?"
In Hinduism, an avatar is the bodily incarnation of a deity on earth. The god can become incarnate in one place at a time as a full avatar or in many places simultaneously through partial avatars called amshas, such that the main form of the god can still communicate with the partial materializations. One could view avatars as embodying the concepts of pantheism (god is all) and polytheism (many gods).
The belief in Hindu avatars is similar to the Christian heresy of Docetism, which is the belief that Jesus Christ only appeared to be human. Docetism teaches that Jesus’ body was spiritual, rather than physical; thus, He was unable to suffer physical pain. In Hinduism, the avatar appears to the devotee in whatever form the worshiper envisions, which, according to Hindu belief could be Mohammed, Krishna, Jesus, Buddha, or any other personal god. An “unqualified” person would take the avatar to be an ordinary human.
The purpose of the avatar’s manifestation is to restore dharma, or righteousness, to the cosmic and social order. Dharma encompasses behaviors such as duty, ritual, law, morality, ethics, good deeds, etc.—anything considered critical to maintaining natural order. That which is unnatural or immoral is called adharma.
Avatars are most often associated with the god Vishnu, one of the members of the Hindu “Great Trinity” or Trimurti (although any Hindu god may manifest as an avatar). Vishnu is considered the maintainer or preserver, as opposed to the other members, Brahma the creator and Shiva the destroyer. According to the Bhagavata Purana, a book of Vedic Sanskrit traditions, Vishnu has incarnated as innumerable avatars in unlimited universes, though there are ten major incarnations, known collectively as Dashavatara.
Some Hindus consider Jesus as an avatar and, more specifically, as the reincarnation of Krishna. However, Jesus was not reincarnated; He was resurrected. Jesus was not an avatar; He is fully human and fully God... After His crucifixion, Jesus was resurrected bodily.
In some ways Jesus may seem to fit into Hindu avatar theism; for example, by bringing the restoration of righteousness, Jesus is, in fact, the only path to eternal salvation. In John 14:6, Jesus said, “I am the way and the truth and the life. No one comes to the Father except through me.” This coming to the Father is accomplished via belief (John 3:18) and repentance (Luke 13:3). The consequences of unbelief are harsh and eternal (Revelation 21:8). First Thessalonians 1:9-10 tells us to turn “from idols to serve the living and true God, and to wait for his Son from heaven, whom he raised from the dead—Jesus, who rescues us from the coming wrath.”