Is there any Christian teaching as to where Adam and Eve were sent after their fall from the Garden of Eden?
The short answer: Not officially anyway, but for some tradition says otherwise!
For example the Catholic Church has no official teachings on this matter. But in the Revelations of Blessed Anne Catherine Emmerich, she mentions that Adam and Eve were expelled to the general region in which because known later as Jerusalem.
I place a caveat here, because even Catholics are not obliged to believe in private revelation, but are permitted to do so.
67 Throughout the ages, there have been so-called "private" revelations, some of which have been recognized by the authority of the Church. They do not belong, however, to the deposit of faith. It is not their role to improve or complete Christ's definitive Revelation, but to help live more fully by it in a certain period of history. Guided by the Magisterium of the Church, the sensus fidelium knows how to discern and welcome in these revelations whatever constitutes an authentic call of Christ or his saints to the Church.
Christian faith cannot accept "revelations" that claim to surpass or correct the Revelation of which Christ is the fulfilment, as is the case in certain nonChristian religions and also in certain recent sects which base themselves on such "revelations".
Catechism of the Catholic Church
It could be stated that it was thanks to the Revelations of Catherine Emmerich that Mary’s house near Ephesus was discovered in 1818. Many of her statements are uncannily accurate.
Now back to the question at hand. In her revelations she states that Adam was expelled from Eden and dwelt in the general area of Jerusalem.
Whilst meditating on the name of Golgotha, Calvary, the place of skulls, borne by the rock upon which Jesus was crucified, I became deeply absorbed in contemplation, and beheld in spirit all ages from the time of Adam to that of Christ, and in this vision the origin of the name was made known to me. I here give all that I remember on this subject.
I saw Adam, after his expulsion from Paradise, weeping in the grotto where Jesus sweated blood and water, on Mount Olivet. I saw how Seth was promised to Eve in the grotto of the manger at Bethlehem, and how she brought him forth in that same grotto. I also saw Eve living in some caverns near Hebron, where the Essenian Monastery of Maspha was afterwards established.
I then beheld the country where Jerusalem was built, as it appeared after the Deluge, and the land was all unsettled, black, stony, and very different from what it had been before. At an immense depth below the rock which constitutes Mount Calvary (which was formed in this spot by the rolling, of the waters), I saw the tomb of Adam and Eve. The head and one rib were wanting to one of the skeletons, and the remaining head was placed within the same skeleton, to which it did not belong. The bones of Adam and Eve had not all been left in this grave, for Noah had some of them with him in the ark, and they were transmitted from generation to generation by the Patriarchs. Noah, and also Abraham, were in the habit, when offering sacrifice, of always laying some of Adam's bones upon the altar, to remind the Almighty of his promise. When Jacob gave Joseph his variegated robe, he at the same time gave him some bones of Adam, to be kept as relics. Joseph always wore them on his bosom, and they were placed with his own bones in the first reliquary which the children of Israel brought out of Egypt I have seen many similar things, bat some I have for. gotten, and the others time fails me to describe.
As regards the origin of the name of Calvary, I here give all I know. I beheld the mountain which bears this name as it was in the time of the Prophet Eliseus. It was not the same then as at the time of our Lords Crucifixion, but was a hill, with many walls and caverns, resembling tombs, upon it. I saw the Prophet Eliseus descend into these caverns, I cannot say whether in reality or only in a vision, and I saw him take out a skull from a stone sepulchre in which bones were resting. Some one who was by his side--I think an angel--said to him, This is the skull of Adam.' The prophet was desirous to take it away, but his companion forbade him. I saw upon the skull some few hairs of a fair colour.
I learned also that the prophet having related what had happened to him, the spot received the name of Calvary. Finally, I saw that the Cross of Jesus was placed vertically over the skull of Adam. I was informed that this spot was the exact centre of the earth; and at the same time I was shown the numbers and measures proper to every country, but I have forgotten them, individually as well as in general. Yet I have seen this centre from above, and as it were from a bird's-eye view. In that way a person sees far more clearly than on a map all the different countries, mountains, deserts, seas, rivers, towns, and even the smallest places, whether distant or near at hand. - The Dolorous Passion of Our Lord Jesus Christ
The Skull at the Foot of the Cross
On some crucifixes a skull and crossbones are shown below the corpus, referring to Golgotha (Calvary), the site at which Jesus was crucified, which the Gospels say means in Hebrew "the place of the skull." Medieval tradition held that it was the burial-place of Adam and Eve, and that the cross of Christ was raised directly over Adam's skull, so many crucifixes manufactured in Catholic countries still show the skull and crossbones below the corpus.
Next, the skull and the crossbones has a dual significance. First, Jesus was crucified just outside the old city of Jerusalem at Golgotha, meaning "Skull Place" in Hebrew. The four Gospels all attest to this fact: Matthew 27:33, Mark 16:22, Luke 23:33, and John 19:17. The word golgotha is an Aramaic form of the Hebrew word gulgoleth, meaning "skull." The Latin word calva, also meaning "skull," is the root for "Calvary."
Secondly, an ancient tradition relates that this spot was also where Adam was buried, hence the depiction of both the skull and crossbones. Today, at the Church of the Holy Sepulcher in Jerusalem, the Greek Orthodox have a chapel built over the rock of Calvary itself, and the place where the cross was erected is marked by a silver disk right below the altar. To the right of the altar, there is a crack in the rock. The Gospel of St. Matthew states that when Christ died on the cross, "the earth quaked, boulders split, tombs opened" (Mt 27:52). The crack continues down to the Chapel of Adam (in the Church of the Holy Sepulcher) where tradition holds Adam was buried and where the precious blood of our Lord dripped upon his bones and his skull. Here the blood of Christ flowing from the Sacred Heart of our Lord would have been a stream of redemption, touching all, even Adam himself. Christ, the new Adam, obedient to the Heavenly Father's will unto to death, conquered the sin committed when the first Adam disobeyed God. The gates of heaven closed by the sin of Adam were now opened by the sacrifice of our Lord.
These two symbols placed on the crucifix help us to remember that our Lord's sacrifice on the cross is the greatest act of love God has shown to us. Therefore, each time we gaze upon our crucified Lord we should be moved to say as did St. Francis of Assisi, "We adore You, oh Christ, and we praise You, for by Your holy cross You have redeemed the world." - Images Surrounding the Crucifix
Some Christians hold to the tradition that Christ was Crucified above the very spot were Adam was buried, now encased in the Basilica of the Church of the Holy Sepulchre. On the ground floor, just underneath the Golgotha chapel, is the Chapel of Adam. According to tradition, Jesus was crucified over the place where Adam's skull was buried. According to some, the blood of Christ ran down the cross and through the rocks to fill Adam's skull. Through a window at the back of the 11th-century apse, the rock of Calvary can be seen with a crack traditionally held to be caused by the earthquake that followed Jesus's death; some scholars claim it is the result of quarrying against a natural flaw in the rock.
During a 1986 repair to the floor of the Calvary Chapel by the art historian George Lavas and architect Theo Mitropoulos, a round slot of 11.5 cm (4.5 in) diameter was discovered in the rock, partly open on one side (Lavas attributes the open side to accidental damage during his repairs); although the dating of the slot is uncertain, and could date to Hadrian's temple of Aphrodite, Lavas suggested that it could have been the site of the crucifixion, as it would be strong enough to hold in place a wooden trunk of up to 2.5 metres (8 ft 2 in) in height (among other things). The same restoration work also revealed a crack running across the surface of the rock, which continues down to the Chapel of Adam; the crack is thought by archaeologists to have been a result of the quarry workmen encountering a flaw in the rock.
Natural stone of Golgotha in the Chapel of Adam below site
Concerning Muslims, they have different traditions:
Some Muslims also believe that Mount Arafat is the place where Adam and Eve reunited on Earth after falling from Heaven, believing the mountain to be the place where they were forgiven, hence giving it the name Jabal ar-Rahmah, meaning 'Mountain of Mercy'. A pillar is erected on top of the mountain to show where this event is believed to have taken place.
Islamic commentators have embellished the Qur'an's account with additional details. Muhammad ibn Jarir al-Tabari wrote that after receiving the breath of God, Adam remained a dry body for 40 days, then gradually came to life from the head downwards, sneezing when he had finished coming to life. 3 The Shiite commentator al-Qummi records the opinion that Eden was not entirely earthly. Having been sent to earth after eating the forbidden fruit, Adam and Eve first arrived at mountain peaks outside Mecca — Adam on Safa, and Eve on Marwa. In this tradition, Adam remained weeping for 40 days, until he repented, at which point God rewarded him by sending down the Kaaba and teaching him to perform the Islamic duty of the Hajj. - Adam and Eve in the Qur'an
The Tomb of Eve, also known as Eve's Grave and Eve's Tomb, is an archeological site located in Jeddah, Saudi Arabia. It is considered by some Muslims to be the burial place of Eve. Prince Faisal, Viceroy of Hejaz, destroyed it in 1928. In 1975, the site was also sealed with concrete by religious authorities, who disapprove of pilgrims praying at tombs.
The tomb of Eve in 1894, during the Ottoman period.
Here follows some Jewish traditions on this matter:
According to traditional Jewish belief, Adam and Eve are buried in the Cave of Machpelah, in Hebron.
Jewish midrashic literature avows that, in addition to the patriarch couples, Adam, the first man, and his wife, Eve, were also interred in the Cave of the Patriarchs, a tradition supported by ancient Samaritan texts. The tradition is supported by the simple wording of Genesis 23:2, which refers to "Kiryat Arba... Hevron" ("arba" means four). Commenting on that passage, Rashi listed the four couples chronologically, starting with Adam and Eve. - Cave of the Patriarchs