Harsh Pant, an worldwide relations professor at King's College London, told AFP that China's backing of Pakistan "has left a very bad taste in India's mouth".
The two leaders have agreed that both countries should work together to deal with the challenge of terrorism, he added.
Prime Minister Modi took the visiting Chinese dignitary to three iconic monuments - Arjuna's Penance, Krishna's Butter Ball and the Shore Temple in Mamallapuram- before settling down for a colourful performance.
The foreign secretary said that President Xi also "welcomed" the investment from India into China including in information technology.
Xi said the two countries should focus on friendship and cooperation, dispel suspicions and doubts, and properly handle differences and sensitive issues.
Leaders like Modi and Xi, who are dominant personalities in their respective systems, use summitry to move the needle, either on trade or security, beyond their sclerotic administrations. "This was the quality time the two leaders spent together one-on-one".
India and China, both in the process of national rejuvenation, are facing many similar challenges, he said, adding that India is willing to deepen and expand pragmatic cooperation with China, and learn from each other in their exchanges.
If Pakistan expected a discussion on Kashmir between Xi and Modi, that did not happen. "We will support China's Fujian province in establishing friendship city relations with Tamil Nadu, Guangzhou and Chennai, and write a new chapter in the history of the maritime silk road", he said. The talks continued on Saturday as well and the leaders of both the nations vowed to jointly combat terrorism and radicalisation, signalling a recalibration of bilateral ties.
A meeting between Xi and Modi in Wuhan, China, in April 2018 was preceded by tensions caused by a 10-week standoff between their countries' armed forces on the Bhutan border. The two countries set up a new high-level mechanism on trade, investment and manufacturing that will have Finance Minister Nirmala Sitharaman on the Indian side and the Chinese vice premier, Hu Chunhua, on the other. Xi and his delegation's attention this weekend was divided between Modi and Washington, where China's chief negotiator, Liu He, met his United States counterparts to achieve a bare-bones trade agreement.
Modi released a short video on Twitter showing him walking barefoot in the sand, collecting the trash in a bag on Saturday morning in Mamallapuram in Tamil Nadu state. It was decided that public of both countries must be brought in the relationship.
"We have seen the report regarding the meeting of Chinese President Xi Jinping with Pakistan's Prime Minister Imran Khan which also refers to their discussions on Kashmir. Today, our Chennai vision will begin a new age in relations between our two countries". Strategic communication between our two countries has also increased. For most part of the last 2000 years, India and China have been economic powers, he said. PM Modi has accepted the invitation.