Jewish manners-Acknowledging the greetings of another:

It is forbidden to ignore the greetings of another, such as to ignore and not respond to another person telling them hello, good morning, Shalom Aleichem.[1] One who does so, and ignores another person’s greetings is called a robber/Gazlan.[2] Not responding to a greeting can cause enmity to spread between the two parties, which is completely avoidable if he would simply acknowledge the greeting. It is for this reason that the Sages required one to greet others, in order to increase love and peace and avoid matters that cause enmity.[3] [Thus, aside for the general social obligation of a human and Jew to act with Derech Eretz and respond to another who greets them, the Sages enacted this as a Halachic requirement, and one who does not do so will be held accountable in his final judgment.[4]]

During learning, prayer, other activities: One must respond to the greetings of another even if he is in the midst of an activity, such as in the middle of work.[5] Even if one is in the midst of prayer he is to respond to another’s greeting[6], unless one is in an area of prayer where he may no longer speak, such as between Baruch Sheamar and Shemoneh Esrei.[7] [Seemingly, one is required to respond even while learning Torah.]

How to respond:[8] One who is greeted with words, such as good morning, Shalom Aleichem, is to likewise respond with words. It does not suffice to simply nod the head.

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[1] Brachos 6b “If he was given Shalom and did not repeat it back to him, he is called a Gazlan” [mentions in many talks of the Rebbe]; Yalkut Shimoni Yeshaya 2:397; Tosafus Bava Kama 73b; see also Tosafus Bava Basra 129b; Sefer Hatanya; Menoras Hamaor Ner 6:3; Raavan Brachos 132; Shibulei Haleket Mila; See Rabbeinu Bechayeh in Kad Hakemach Shalom; Chesed Lealafim 156:8; So is also evident from Mishneh Brachos 13a, Tur, Michaber and Admur 66:1 that one may answer Shalom even during Shema due to Darkei Shalom

[2] Brachos ibid

The reason: As the verse [Yeshayahu 3:14] states “Veatem Biartem Hakerem Gzeilas Heani Bivateichem” [Brachos ibid] This refers even to an Ashir, however, the Pasuk speaks of a poor person as by a poor person there is nothing to steel but his greeting [and hence that is how we know that even not returning a greeting is considered stealing]. [See Rashi ibid and Kol Eliyahu of Gr”a ibid] The reason why this is considered steeling is because one has taken the blessing that he received from the person and did not give it back. [See Hatanya ibid]

[3] Menoras Hamaor Ner 6:3; Kad Hakemach ibid; Admur ibid “One may answer due to Darkei Shalom”

[4] See Chagiga 5a “On all the acts Hashem will bring to judgment, on all concealed matters, Shmuel says that this verse refers to one who spits in front of his friend and it became repulsive before him”;

[5] Tosafus Bava Kama 73b; see also Tosafus Bava Basra 129b

[6] Mishneh Brachos 13a, Tur, Michaber and Admur 66:1;

[7] From the letter of the law this is allowed until the point of Shemoneh Esrei, as stated in Admur, Michaber, Tur and Brachos ibid, however, the custom today is not to answer once one is past Baruch Sheamar, as people are no longer Makpid if someone does not answer them while they are in the midst of Davening. [See Admur 66:1 “If the person is not Makpid don’t answer him”; M”A 66:2 in name of Rishonim; Chinuch Mitzvah 420; Elya Raba 66:2; M”B 66:2 that so is the custom]

[8] Chesed Lealafim 156:8