Gengis Khan conquered more land than anyone else but after looking at a map of the Mongolian empire I noticed he never invaded India
Why might he have chosen not to do so?
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Genghis Khan mostly made a one-way trip. There were two main branches of the Silk Road (which wasn't an actual road, but rather an itinerary). One passed north of the Himalayas and one south (or by ship through the Arabian Sea). These were the easiest customary itineraries one could take to cross Asia, as they minimized the amount of mountain-crossing one had to do.
Mountain chains were perilous, cold, uninhabited, barren and rather inaccessible landscape features, so it is self-evident why he would try to avoid marching an enormous army with lots of horses (that need lots of feed) through a place with little support capacity and no loot.
So that explains why he had to choose one of the two roads, but it doesn't explain why he chose to cross the steppes, forests and deserts of Mongolia, Siberia, "the 'stans" (central Asia), Russia and so forth. Well there are a number of reasons why he might have done so.
The more practical one considers logistics. When he conquered China, he didn't personally oversee the conquest of most of the territories displayed in your map. The mongol conquest of china (the yuan domination of China) wasn't completed until some decades after his death (he died 1227, your map is 50 years later). In his first campaigns as Great Khan, Genghis led some successful raids mainly in northern China, conquering the Jin and Xia dynasties (but not the Song), and making Korea a vassal state. Now logistics (how you organize and move your armies) gets important: after each of the initial raids, which were conducted in mongol fashion, he retreated his main army back to his home land. But after having subdued northern china, he had to look for new targets to invade so he chose the next closest target, which was the Kara-Khitan Khanate. After conquering them, he couldn't retreat, if he wanted to keep conquering, because he would have locked himself in: conquered land in the south and south east, mountains, deserts and a conquered khanate in the west and wasteland to the north. So instead he chose to keep going west, where there were still lands to feed his army off and cities ripe with booty (the kind of booty you can plunder). So he defeated the Khwarezmids the Bulgars, Crimea (now, see? everyone wants that.) and the Kievan Rus (not conquering them though, that work was mostly done after his death, by his generals and succesors). After the Russian campaign, he did eventually return home. There, while trying to crush some revolts, he died, therefore ending any plans of an Indian campaign he might have had.
The second reason i want to mention, is a much simpler one. Consider the ethnic and cultural distribution of that time. Mongol and Turkic tribes had similar cultures. They were both nomadic people who lived in steppes, bred horses and were tengri. The "mongol" tribes Genghis united, were probably neither ethnically nor linguistically purely mongol, but rather culturally. The united tribes comprised Mongol as well as Turkic tribes, that spoke other languages and might have had other genetics (i don't know), but had ultimately more in common among them than they had with the rest of their known world (Chinese, the Rus and the Sultanates were partly Urbanized, mostly sedentary, agricultural monarchies). Also the vast majority of the inhabited part of central and northern Asia were vast, meager areas not suited for agriculture, that were inhabited by above mentioned Turkic and Mongol tribes. So in the end the land north of the Himalayas looked very familiar to Genghis and his people. It was similar to their home, it was governed by the same rules of nature and most importantly it was inhabited basically by family (the kind of family you go to war against). Mongolia was connected to Europe by a net of (real and imagined) familiar relationships and close customs, more so than it was connected to the rest of Asia...easier to conquer, keep and traverse. Maybe, the reason why Genghis crossed Asia north of the Himalayas, was that it just felt more like home.
India's climate is the diametric opposite of Mongolia's climate. That is hot and wet versus cold and dry. Genghis Khan's troops would have suffered horribly (both horses and men) in most parts of India. The humid air was a real "dampener" (pun intended) for Mongol bows.
The parts of "India" that the Mongols occupied, (Pakistan and Kashmir per a map in the question), were higher in altitude, cooler, and less humid than the rest of (modern) India, i.e. closer to the Mongol climate.
This actually depends on how you define “India”. If you mean “the territory of the present-day Indian Republic” then yes, Genghis did not conquer any significant portion of “India”. But do not forget that in pre-modern times “India” included present-day Pakistan. As you can see from your map, Genghis did conquer a large part of that territory.