During a twelve-day span in November, Barack Obama met with the leaders of Japan, China, and India. The substance of his trip to the Far East was eclipsed by an imprudent bow before the Japanese emperor. Obama’s White House state dinner for Indian prime minister Manmohan Singh was overshadowed by a pair of party-crashers. Yet the tabloid-friendly news items should not distract us from the profound significance of U.S. dealings with Asia’s Big Three.

“This is the first time in history when there have been three powerful countries in Asia, all at the same time,” notes former Economist editor Bill Emmott in his 2008 book, Rivals. “Managing the relationship between China, India and Japan promises to be one of the most important tasks in global affairs during the next decade and beyond, comparable in importance to the need to find peaceful ways to manage the relationships between Europe’s great powers during the twentieth century.”

As President Obama tackles this challenge, he should seek to preserve and build upon the legacy of George W. Bush, whose Asia policies were far more successful than his myriad critics have acknowledged. Indeed, the Bush administration quietly pulled off an impressive diplomatic feat: strengthening U.S. ties with China while concurrently boosting America’s partnerships with India, Japan, Australia, Indonesia, Vietnam, and other Asian-Pacific countries wary of China’s growing influence.

Writing in Foreign Affairs shortly before the 2008 U.S. presidential election, Japanese journalist Yoichi Funabashi, editor-in-chief of the Asahi Shimbun newspaper (a leading daily), lauded Bush for “promoting sound relationships with Beijing and Tokyo simultaneously.” His administration elevated the U.S.-Japan alliance “to an almost unprecedented level,” said Funabashi, even as it solidified “a healthy working rapport” with China. In addition, Bush embraced India and signed a groundbreaking bilateral nuclear pact, giving the South Asian giant crucial assistance in its drive for global-power status. Last December, Harvard historian and India expert Sugata Bose told me that bolstering U.S. relations with the world’s largest democracy “may turn out to be the most significant foreign-policy achievement of the Bush administration.”

Such achievements conflict with the popular storyline that Bush ignored Asia after 9/11. In fact, a 2008 survey by the Chicago Council on Global Affairs (CCGA) and the East Asia Institute (EAI) found that pluralities in Japan (47 percent), China (45 percent), and South Korea (42 percent) — plus a majority in Indonesia (58 percent) — felt U.S. influence in the region had increased over the past decade, whereas only 28 percent of Chinese, 24 percent of South Koreans, 20 percent of Japanese, and 11 percent of Indonesians thought U.S. influence had declined. Meanwhile, 76 percent of Vietnamese, 69 percent of Japanese, 66 percent of Chinese, and 54 percent of South Koreans described America’s influence in Asia as “somewhat” or “very” positive.

Bush’s first serious Asia test was the April 2001 EP-3 spy-plane dispute with China. That same month, he approved a huge arms sale to Taiwan and made a provocative declaration: Speaking to ABC’s Good Morning America, Bush pledged that if China ever attacked the island democracy, the U.S. would do “whatever it took to help Taiwan defend herself.” This remark triggered a media frenzy, but the White House insisted that U.S. policy toward Taiwan had not changed. In December 2003, Bush publicly chided then–Taiwanese president Chen Shui-bian for a series of “comments and actions” that appealed to pro-independence sentiment on the island and, from a U.S. perspective, needlessly riled Beijing. Bush affirmed that Washington was opposed to unilateral disruptions of the China-Taiwan status quo.

The rebuke of Chen, which angered many conservatives, reflected the deeper pragmatism that guided Bush’s China policy. Though he prodded Beijing on human rights, met with Chinese dissidents and the Dalai Lama, and praised Taiwan as “a free and democratic Chinese society,” Bush’s approach to the Middle Kingdom was fundamentally realist. His administration adopted some foolish protectionist measures that irked the Chinese, but it also created the U.S.-China Strategic Economic Dialogue, which was designed to manage what former Treasury secretary Hank Paulson called “the most important bilateral economic relationship in the world today.”

Yet the Bush administration recognized that engagement with Beijing should be balanced by greater cooperation with the Asian-Pacific democracies. While Tony Blair was his most conspicuous chum among foreign statesmen, Bush also forged warm personal bonds with Japan’s Junichiro Koizumi (prime minister from 2001 to 2006), Australia’s John Howard (prime minister from 1996 to 2007), and India’s Singh (prime minister since 2004). Their four countries — America, Japan, Australia, and India — coordinated relief efforts after the 2004 Asian tsunami, and in September 2007 they held large-scale naval exercises in the Indian Ocean (along with Singapore).

Though he was frequently caricatured as a pro-democracy crusader, Bush also pursued closer ties with undemocratic Vietnam, a country with a young, energetic, and broadly pro-American population of more than 86 million, whose leaders are eager to cultivate friendly relations with the U.S. as a hedge against China. During the Bush years, notes a 2009 Congressional Research Service report, “the United States and Vietnam dramatically upgraded diplomatic and strategic aspects of their relationship to the point where the two countries have all-but-normalized bilateral relations, at least from the U.S. point of view.” In 2007, they signed a trade and investment framework agreement. A year later, they launched official “political-military talks,” started negotiations on a bilateral investment treaty, and established the U.S.-Vietnam Education Task Force.

U.S. trade with Vietnam flourished under Bush, who also completed bilateral free-trade pacts with Singapore, Australia, and South Korea. The Korea deal, signed way back in June 2007, still awaits congressional approval. Obama should pressure Hill Democrats to quit their stonewalling. There is no good reason for Congress to alienate the pro-American government of South Korean president Lee Myung-bak.

Obama should also learn from his predecessor’s mistakes with the other Korea. Bush’s second-term decision to ease sanctions against the North and remove it from the State Department’s list of terror sponsors exasperated Japan and failed to persuade Pyongyang to scrap its nuclear-weapons program; indeed, North Korea conducted its second nuclear test this past May. Obama should aggressively enforce United Nations Resolution 1874, which authorized new sanctions and called for inspections of North Korean cargo. As former National Security Council (NSC) official Victor Cha points out in the Washington Quarterly, “The resolution effectively institutionalizes, in a U.N. context,” Bush’s Proliferation Security Initiative, “making it both more effective and more inclusive.” Cha also recommends that the U.S. prepare thoroughly for a possible collapse of the North Korean regime, and that China be included in the relevant multilateral discussions.

Of course, given China’s massive holdings of U.S. government debt, Washington has limited leverage over Beijing. In Superfusion, his new book on U.S.-China economic interdependence, economist Zachary Karabell argues that in the aftermath of the global financial crisis, this interdependence “has become even more vital to the prosperity not just of hundreds of millions of Americans and more than a billion Chinese but for everyone everywhere.” As we’ve heard countless times since the crisis erupted, the U.S.-China relationship must be rebalanced: Chinese need to spend more, and Americans need to save more; China needs to expand its domestic consumer market, and America needs to produce more exports.

While the U.S. and China must collaborate on a range of issues — economic rebalancing, trade, currency, energy development — Obama should eschew the concept of a U.S.-China “G2,” which causes heartburn in New Delhi, Tokyo, and elsewhere. The president should remember that, in the 2008 CCGA/EAI survey, only 27 percent of Indonesians, 21 percent of South Koreans, and 10 percent of Japanese said they would be comfortable with China’s becoming the “leader of Asia.” In general, Obama should avoid fostering the idea that America’s relationship with China is being elevated at the expense of its partnerships with India, Japan, and others.

Take the U.S.-China Joint Statement released on November 17. Indian officials were troubled by a passage that suggested the U.S. and China were “ready to strengthen communication, dialogue and cooperation on issues related to South Asia and work together to promote peace, stability and development in that region.” The notion of China’s brokering diplomatic negotiations between India and Pakistan is anathema to New Delhi.

Sometime over the new few decades, India will surpass China to become the most populous country on earth. Goldman Sachs has projected that India could have a bigger economy than the United States by 2050. After neglecting New Delhi for much of his first year in office, Obama used Prime Minister Singh’s recent visit as an opportunity to tout the bilateral partnership that Bush did so much to nurture. The Indians want to see a long-term U.S. commitment to pacifying and stabilizing Afghanistan. They also want U.S. authorities to exert more pressure on Pakistan. “They feel the source of the problem in Afghanistan is the Pakistani army,” says C. Raja Mohan, a prominent Indian strategic analyst.

In Japan, meanwhile, a new prime minister from an inexperienced party is grappling with deflation, surging public debt, an aging workforce, and a shrinking population. Yukio Hatoyama, who took office in September, following the country’s most severe recession of the post-war era, inherited a slew of economic woes. He immediately spawned a new foreign-policy dilemma by wrangling with the Obama administration over the planned relocation of U.S. military personnel in Japan. Hatoyama has advocated a more “equal” U.S.-Japan alliance.

Handling the somewhat disorganized and amateurish Hatoyama government will require diplomatic finesse. Luckily, Obama’s Asia team is “every bit as good” as the Bush team, says Michael Green, who served as a high-level NSC official for Asia from 2001 to 2005. Obama’s seasoned and well-respected Asia hands include Kurt Campbell (the top State Department official for East Asia) and Jeffrey Bader (senior director for Asian affairs at the NSC).

Emmott, the former Economist editor, stresses that, for all the attention devoted to the Middle East since 9/11, “the most important long-term trend in world affairs does indeed remain the shift in economic and political power to Asia.” Bush’s skillful diplomacy both adapted to and influenced this shift. Outside of North Korea, it yielded salutary and even historic results, which the Obama administration can now harvest.