Nestorianism, is a heresy advanced by Nestorius (d. 451?), patriarch of Constantinople. It declared that Jesus was two distinct persons, one human and one divine. Nestorius opposed the title of Mother of God for the Virgin Mary, contending that she was mother of the human person of Jesus. The councils of Ephesus (431) and Chalcedon (451) clarified the orthodox Catholic view that Jesus' two natures are inseparably joined in one person and partake of the one divine substance. Elements of Nestorianism survive in the modern Assyrian Church, based mainly in the Middle East; its members number c. 400,000. Since the Thomas Christians of Malabar (India) were ruled by the Chaldean Church from 7th century until 16th century, many people, especially the Portuguese suspected Nestorianism in the Malabar Church. From what many eminent historians say, we can conclude that probably they were influenced by the heresy, but because of the type of relationship they had with the Chaldean Church, they were not in schism. For more details, see Early Christians of India and Catholicism.
Monophysitism, which means "belief in one nature", is a heresy that developed in the 5th century as a reaction against Nestorianism which also developed in the same century. Monophysitism is a doctrine that in the person of Jesus there was but a single nature which was divine nature. This doctrine was condemned at the Council of Chalcedon (451). For invalidating Chalcedon, the East was put under excommunication by the pope until 519. In Syria, Egypt, and Armenia, monophysitism dominated, and a permanent schism set in by 600, resulting in the creation of the Jacobite, Coptic, and Armenian Churches. It continued to be followed as a doctrine in the northern part of the Antiochene Patriarchate (West Syrian), by nearly the entire Church of Alexandria (and later of Ethiopia), by Armenia, and, in the 16th century, by a portion of the St. Thomas Christians in India. National prejudices of Syrians and Egyptians against the Byzantines were a major factor in the adherence to and extension of Monophysitism in the early Byzantine period. The Byzantine emperors tried to eradicate Monophysitism from their empire in an effort to achieve civil and religious unity, but Empress Theodora, wife of Justinian I, promoted its spread throughout all of Syria, Mesopotamia, and other countries by sending Jacob Baradai into Syria to consecrate Monophysite bishops and to secure the foundation of the Jacobite Syrian Church.
The Syrian Monophysites are called Jacobites after Jacob Baradai (d. 578), who, during the persecutions waged by Justinian I against Monophysitism, secretly consecrated 27 bishops and some 2,000 priests, thus giving a strong hierarchy to the Syrian Monophysite Church. The modern Jacobite patriarch claims the ancient see of the Patriarchate of Antioch as his legitimate see, and resides in Damascus.
Jacobites of India: Owing to enforced Latinization by the Portuguese, who were the civil and ecclesiastical authorities from the 17th century in Malabar (presently Kerala), India, a large number of Thomas Christians of the Chaldean rite embraced the Jacobite Church under the Monophysite patriarch of Antioch.
For more details, see The Orthodox Churches in India.