CARBIS BAY, England — Addressing U.S. troops shortly after his arrival in England this past week, President Biden took pains to stress the importance of working with allied nations, emphasizing a partnership "grounded on democratic ideals and a shared vision of the future."

He underscored his belief in the importance of Article 5, the NATO agreement that an attack on one nation is an attack on all, calling the U.S. commitment “rock solid” to the alliance’s “sacred obligation.”

And he took a stern posture ahead of a planned meeting with Russian President Vladimir Putin — warning that he planned to “let him know what I want him to know.”

Through it all, Biden never once mentioned the name of his predecessor, Donald Trump. Yet Trump’s shadow has loomed large over Biden’s first trip abroad as president — an eight-day swing through Cornwall, England; Brussels; and Geneva, where Biden is being welcomed as much for who he is not as for who he is.

The global specter of Trump has transformed itself from an active storm to a threatening cloud, creating an overseas atmosphere in which Biden was greeted with equal parts optimism and skepticism. One senior European official described his horror at the Jan. 6 assault by pro-Trump rioters on the U.S. Capitol — and said he had an even worse feeling reading opinion polls since then showing that a substantial portion of Republican voters believe Biden is an illegitimate president, a baseless claim perpetuated by Trump.

“Your democracy is in serious trouble,” the official said, speaking on the condition of anonymity to talk frankly about an ally.

Biden, by contrast, was greeted with delight by leaders at the Group of Seven gathering of the world’s wealthy market democracies, who are relieved that Trump’s tantrums will be replaced by Biden’s backslapping.

Speaking from a beach in Cornwall, British Prime Minister Boris Johnson called working with Biden a “breath of fresh air.”

President Biden met with French President Emmanuel Macron on June 12 in a bilateral meeting during the annual G-7 summit. (Reuters)

And on Saturday, sitting next to Biden in Carbis Bay — the aquamarine water behind the leaders looking as crystalline as the day’s sky — French President Emmanuel Macron praised the U.S. president for his global efforts on combating the coronavirus pandemic and climate change, saying: “It’s great to have a U.S. president part of the club and very willing to cooperate.”

Leaders have dark memories of past NATO summits, when Trump threatened to pull the United States out of the alliance — and once shoved the prime minister of Montenegro during a photo op. They remember the G-7 meetings where Trump repeatedly interrupted discussions to ask why they couldn’t invite Putin, a leader the rest of them viewed as an adversary but whom Trump saw as a friend.

At the 2018 G-7 summit in Quebec, Trump reportedly threw two Starburst candies toward German Chancellor Angela Merkel, refused to sign the final joint statement with U.S. allies and insulted the gathering’s host, Canadian Prime Minister Justin Trudeau, as “very dishonest & weak” in a tweet — all before jetting off to a meeting with North Korean leader Kim Jong Un.

European allies were so embittered by their encounters with Trump that they rejected his efforts to host an in-person G-7 summit last year as the pandemic was raging.

“It’s good that the U.S. are back and it’s good that the G-7 is back, too,” European Commission President Ursula von der Leyen said Thursday.

But below the surface-level bonhomie, Trump’s legacy continues to shape European attitudes toward Washington. Four years ago, few on the continent expected Trump to so quickly sweep away decades of American orthodoxy in his relations with allies. Now, few are confident that Biden won’t be replaced by Trump — or a Trump-like figure — in 3½ years.

“We are facing major democratic issues on both sides,” said Pierre Vimont, a former French ambassador to Washington. “Whether Trump will come back is a major concern for the Europeans.”

Ian Bremmer, president of the Eurasia Group, a global risk consultancy, said: “The idea that Biden becomes president and can say, ‘America is back,’ is not remotely coherent to allies around the world.”

Bremmer pointed both to drastic actions Trump took during his four years — such as withdrawing from the Iran nuclear deal and pulling out of the Paris climate agreement — and to the still-fractured domestic political climate in the United States as reasons for European apprehension.

“They saw what happened on January 6th, they saw the second impeachment effort after that,” Bremmer said. “They see where the U.S. is going right now, in terms of all of the efforts to screw with electoral process in different states across the country, the support for a strongly Trumpian figure in the Republican Party . . . and the support for an increasingly vocal and progressive left in the Democratic Party.”

In part because of those concerns, European leaders have indicated that they are unlikely to fall in lockstep behind all of Biden’s initiatives. At this year’s virtual Munich Security Conference — an annual gathering that has traditionally been a forum for transatlantic togetherness — Macron and Merkel both declared their relief at Biden’s arrival but said their obligation was to act in Europe’s interests.

In the view of some, what has happened in recent months has confirmed that strategy. Biden’s decision to withdraw from Afghanistan was done with little consultation with the NATO allies who also have troops there, which led to grumbling inside the alliance that the move fit into a Trumpian mold. Biden has promised a “foreign policy for the middle class,” but Europeans wonder whether that might simply mean Trump-era trade barriers in a more genteel wrapping.

“What does it mean for the trade front? Is this protectionism in another cloak, or is this something that the Europeans can relate to?” said Rosa Balfour, director of Carnegie Europe, the Brussels office of the Carnegie Endowment for International Peace.

NATO Secretary General Jens Stoltenberg, a polished diplomat who sought ways to work with Trump, declined to compare the two presidents. But, he said, “the lesson learned, in uncertain times, is that institutions like NATO” are stronger than the individuals that come and go in the governments of its members.

“Strong institutions are able to weather storms and difference,” Stoltenberg said. “If anything, the last four years have proven [NATO] is a robust institution.”

Matthew P. Goodman, senior vice president for economics at the Center for Strategic and International Studies, said that to reassure the Europeans, “Biden has to show he’s confident, that the U.S. is back, that the U.S. is ready to do what it needs to do domestically.”

“We will not slide back into Trumpism — that’s the message he needs to try to convey, and that’s a tough message, especially this early into his administration,” Goodman said.

Appearing with Macron on Saturday, Biden gestured to the French leader when asked by reporters if America is back.

“Yeah, definitely,” Macron responded.

On Thursday, Biden officially announced that the United States is purchasing 500 million Pfizer-BioNTech coronavirus vaccine doses to donate to the rest of the world — representing half of the 1 billion vaccine doses that G-7 leaders committed to providing to vulnerable populations around the globe.

“In times of trouble, Americans reach out to offer help and to offer a helping hand,” Biden said. “That’s who we are.”

Implicit in his announcement was the idea that the United States has the coronavirus increasingly under control domestically, and is now ready to assist the rest of the world.

When it comes to his Wednesday summit in Geneva with Putin — expected to be the thorniest part of his trip — Biden has already distinguished himself from Trump, who seemed intent on forging a friendship with the authoritarian leader and who accepted Putin’s word over his own intelligence team’s.

“He’s certainly not going to want to be his friend, like President Trump did,” said Michael McFaul, who served as ambassador to Russia under President Barack Obama. “That’s easy — he’s over that hurdle already.”

So far, Biden’s approach on the world stage seems to be improving views of America. A Pew Research Center global survey released Thursday found that among adults across 12 countries surveyed, 62 percent now have favorable ratings of the United States, compared with 63 percent who felt unfavorably last year. The survey also found that a median of 75 percent across the countries had confidence in Biden to “do the right thing regarding world affairs,” compared with 17 percent who had confidence in Trump last year.

Small minorities view Biden as arrogant and dangerous, compared with majorities who said Trump was each of those things four years ago, while majorities see Biden as more qualified and as a stronger leader, according to the survey.

Yet even in exile, Trump seems determined to complicate Biden’s visit. In a statement Thursday, the former president touted his own 2018 summit with Putin in Helsinki as “great and very productive,” before sending Biden off with faux salutations.

“Good luck to Biden in dealing with President Putin—don’t fall asleep during the meeting, and please give him my warmest regards!” Trump wrote.

Michael Birnbaum reported from Riga, Latvia. Karen DeYoung in Washington and Anne Gearan in Plymouth, England, contributed to this report.