Ukraine pressed its advance into Russia for a fourth day on Friday, battling to capture a town near the border and sending small units to conduct raids further into the western Russian region of Kursk, independent military experts and pro-Kremlin military bloggers said.

As Ukraine tried to capitalize on its surprise incursion, the Russian military announced it was sending more troops and armored vehicles to try to repel the attack. Russian television released videos of columns of military trucks carrying artillery pieces, heavy machine guns and tanks.

At the same time, the Ukrainian authorities were evacuating 20,000 people from the Sumy region, which sits across the border from Kursk, in what appeared to be anticipation for Russian retaliatory strikes.

Military analysts say the Ukrainian assault is the largest on Russian soil since the war began. The offensive, which began on Tuesday, has temporarily shifted the focus of the war, opening a new front inside Russia and prompting Moscow to scramble to halt the Ukrainian advance.

But the operation has raised questions about whether it is worth the risk, given that Ukrainian forces are already stretched. It is also not clear whether the mission will help Ukraine improve its position on the rest of the battlefield, where it has been steadily losing ground for many months.

Kyiv’s allies in the past have been wary of Ukrainian incursions in Russia, fearing that it could escalate the war, but there have been no public indications from Western capitals that they oppose the assault. The United States has said that the Ukrainian incursion does not violate American guidance.

However, senior American officials have said privately that they did not get a heads-up about the operation and were still seeking clarity about its logic and rationale.

The officials said they understood Kyiv’s need to change the optics and the narrative of the war, but that they were skeptical that Ukraine could hold the territory long enough to force Russia to divert significant forces from the offensives it is pressing in eastern and southern Ukraine.

“It’s a gamble,” said one senior administration official.

Still, Mykhailo Podolyak, a top presidential adviser, was upbeat about the international response. “Most quietly approve,” he wrote on social media on Thursday evening, adding that a significant part of the world now considers Russia “a legitimate target for any operations and types of weapons.”

The fighting showed no signs of abating on Friday, with the Ukrainian military saying that it had struck a Russian airfield in the Lipetsk region, which borders Kursk, hitting warehouses that contain guided aerial bombs. Local Russian authorities said a large drone attack had caused several explosions and that a fire had broken out at a military airfield.

The Ukrainian authorities also said a Russian strike on a supermarket in Kostiantynivka, an eastern town 200 miles south of the area of the fighting, killed 14 people and wounded 43 others. The claims from both sides could not be independently verified.

The Ukrainian military has enforced a policy of silence about the operation, and has not publicly acknowledged launching a cross-border attack.

Military analysts said the attack had involved elements of at least four brigades in a rare example of successful maneuver operations involving support from artillery, air defenses and electronic warfare, resulting in quick advances on the ground.

“It seems to be a fairly well-coordinated an planned combined armed operation,” said Franz-Stefan Gady, a Vienna-based military analyst. “You have electronic warfare assets that were deployed to jam Russian command and control. You have air defenses that were moved in to create air defense bubbles around the Ukrainian advance. And then you have fairly effective mechanized formations moving forward at a fairly steady pace.”

Mr. Gady and other experts said the main question now is whether Ukraine can maintain the momentum and turn the success on Russian territory into useful gains. The Ukrainian Army has few reserves it can pour into the fight, and it continues to suffer from shortages of weapons and ammunition, analysts say.

It also remains unclear what Ukraine ultimately hopes to accomplish. A senior Ukrainian official who spoke the on condition of anonymity to discuss the operation said the goal was to draw Russian troops away from other parts of the front line where Ukrainian units are struggling. But military experts said that Russia would likely be able to respond with reserves who were not fighting in Ukraine.

“Does it really solve any of the larger military strategic problems that the other parts of the front line are suffering from?” Mr. Gady asked.

A map of the battlefield by the Black Bird Group, a Finland-based organization that analyzes images from the battlefield, shows that Ukrainian troops have gained about 100 square miles of Russian territory since the beginning of the attack, although it remains unclear whether they have secured control of all of it. They have advanced past two lines of Russian defenses.

In particular, the Ukrainian army has closed in on Sudzha, a small town of about 6,000 people six miles from the Ukrainian-Russian border.

Emil Kastehelmi, an analyst from the Black Bird Group, wrote on social media that some Ukrainian units appeared to be conducting probing raids further north in the direction Lgov, a town about 50 miles from the border, in what appears to be a test of Russian defenses.

A video posted on social media on Friday morning and verified by The New York Times showed a column of destroyed Russian military vehicles just east of Rylsk, a town west of the border area captured by Ukraine.

It remains to be seen whether Ukraine will try to push further into Russian territory to solidify control over the area it has captured, or retreat after a few days, as has happened in previous, smaller-scale cross-border raids.

Mr. Kastehelmi said Ukraine could not continue further north without widening its flanks and exposing itself to Russian counterattacks. “Time is also running against Ukrainians,” he wrote. “Russians won’t be disorganized forever.”

Eric Schmitt contributed reporting from Washington, D.C.

On the day that Hamas’s political leader was assassinated in Iran, small groups of Palestinians in a number of West Bank cities turned out to protest, some chanting pro-Hamas slogans and waving the armed group’s green flag.

Mahmoud Abbas, the president of the Palestinian Authority, which administers parts of the Israeli-occupied West Bank, condemned the July 31 killing of the leader, Ismail Haniyeh. The Palestinian Authority, a political adversary of Hamas, ordered flags to be lowered to half-staff and called for a day of strikes and business closures, while a wake for Mr. Haniyeh drew political leaders from across the West Bank.

This outpouring of sympathy was notable because unlike Gaza, which Hamas has controlled for most of the past two decades, the West Bank and the Palestinian Authority are dominated by Hamas’s main rival, the more moderate Fatah faction. And the Palestinian Authority generally has shown little tolerance for such open shows of support for armed groups in the past, at times using force to break them up.

In the 10 months since the Oct. 7 Hamas-led attack on Israel, the Palestinian Authority has been losing support to factions like Hamas that favor armed struggle and are actively fighting Israel, according to a recent poll by the Palestinian Center for Policy and Survey Research. At the same time, deadly Israeli raids and attacks by Jewish settlers on Palestinians in the West Bank have escalated.

Israeli officials say those raids are aimed at preventing a second front from opening up in the West Bank while the war in Gaza is ongoing. Israel also accuses some of the armed groups in the West Bank of plotting attacks against it.

“The P.A. is reading the room right now,” said Tahani Mustafa, a senior Palestine analyst at the International Crisis Group. “If they were to clamp down on Hamas supporters, it would be absolutely disastrous,” she added.

“The P.A. recognizes that it is deeply unpopular, which has been evidenced by opinion poll after opinion poll, especially since the 7th of October,” Ms. Mustafa said, adding that repressing shows of support for Hamas during a period of mourning for a leader who is arguably more popular than Mr. Abbas “would be political suicide.”

The hands-off approach to the demonstrations last week signaled a political balancing act by the Palestinian Authority, which has suffered from low approval ratings and a crisis of legitimacy while Hamas — designated as a terrorist group by the United States and Israel — has gained supporters.

Mr. Haniyeh was assassinated last week in the guesthouse where he was staying in Tehran, where he was visiting to attend the inauguration of the new Iranian president. Iranian officials and Iran-backed Hamas blamed Israel, an assessment also reached by several U.S. officials. Israel has not publicly taken responsibility.

“Assassinating Palestinian political leaders is something the Palestinian Authority is going to speak out about,” said Diana Buttu, a lawyer and former legal adviser to the Palestine Liberation Organization, which represents Palestinians internationally and is also dominated by Fatah.

Tolerating expressions of sympathy for Hamas “is a way of allowing people to express sentiment and let out anger,” she added. “But also, I genuinely think this is something that saddens them. It is part of Israel’s history of assassinating our leaders.”

The political chasm between Hamas and the Palestinian Authority has divided Palestinians for nearly two decades as each has tried to position itself as the legitimate leader. In 2006, Fatah lost a legislative election to Hamas. The next year, Hamas fighters routed Fatah security forces from the Gaza Strip and forcibly seized control of the territory.

Over the years since then, multiple efforts to reconcile the rival factions have failed.

But last month, in an unusual show of unity, Fatah and Hamas signed a joint statement in Beijing. And though it is clear that the prolonged Gaza war and questions over who will govern postwar Gaza have made Palestinian unity even more urgent, there are few other signs that the factions are actually bridging their differences.

The joint statement, which was also signed by other smaller Palestinian factions, supports the formation of a temporary government for Gaza and the West Bank, and said the new government should begin working on uniting Palestinian institutions in both territories, reconstructing Gaza and preparing for national elections.

The Biden administration has said repeatedly that a revamped Palestinian Authority should play a role in postwar Gaza.

But even those Palestinians who saw the development in Beijing with a measure of hope have low expectations, as previous attempts to broker unity have also resulted in joint statements and agreements without any lasting progress.

Some Palestinians have long been critical of the Palestinian Authority and its security forces, which have quietly helped Israeli intelligence agencies target Palestinians accused by Israel of militant activity, including Hamas members.

The critics argue that these institutions have become little more than subcontractors for an occupying power, exerting authoritarian control and, at times, violently cracking down on dissent.

Ms. Mustafa said the somber mood among many Palestinians in the West Bank on the day Mr. Haniyeh was killed was evidence of the growing support for Hamas.

In a video of one protest in the West Bank city of Jenin on the night of July 31, one Palestinian man yelled, “We from the land of Jenin affirm that we are all Hamas,” as he led dozens of people walking through the streets.

“In terms of Hamas’ popularity, yes they are the de facto leaders of Palestinians, whether we like it or not,” Ms. Mustafa said. “They are the only ones fighting for Palestinians in the face of no international protections.”

The residents of the southeast Liverpool neighborhood of Edge Hill had spent Wednesday preparing for trouble.

Parents were called to pick up children early from nursery school. Shop owners pulled their shutters down over glass storefronts. And in the semidetached brick houses on and around Overbury Street, where generations of the same families have lived alongside newer arrivals, locals pulled their curtains as evening approached.

What they feared was another night of the anti-immigrant violence that had rocked the country in the week since a deadly stabbing attack nearby in Southport that was falsely rumored as being carried out by a migrant.

What they got, instead, was a night of near celebration by people opposed to the racism and anti-immigrant sentiments that drove the week of rioting in cities and towns across Britain.

People in Liverpool had been especially unnerved since an online list of what were said to be new far-right targets for protests included a local charity that works with asylum seekers. Neighbors texted neighbors to head to the streets to counter any racist rioters. Local unions and leaders of neighborhood mosques also put out the word, as did a nationwide collective called “Stand Up to Racism.”

So as helicopters circled overhead on Wednesday night, and police officers on horseback patrolled the streets, young women handed out snacks and water bottles in front of the boarded-up windows of the targeted charity. Another group set up a makeshift first aid area across the street in case of emergency, given the unbridled violence of the past riots. And a white-haired man with a long beard propped a megaphone next to a speaker on his metal walker and played peace songs.

People carried signs reading “Not in our city,” and “Will trade racists for refugees.”

“They all had one thing in mind; it was to not let this hate get a foothold,” said Ewan Roberts, who manages Asylum Link Merseyside, the charity that was on the target list.

And then, the far right was a no-show.

In some ways, the gathering of hundreds of antiracism demonstrators was not unexpected in Liverpool, a multicultural city with proud working-class roots.

But similar protests were staged in cities across England on Wednesday night as thousands of people angered by the earlier violence decided to make their voices heard. That violence had included rioters trying to set fire to a hotel in the city of Rotherham while asylum seekers and other guests were inside. Some rioters pummeled police officers so hard they had to go to the hospital. A fire was set in a community library on the northern outskirts of Liverpool over the weekend.

Some of the Liverpool residents who turned out in force Wednesday were especially angry that what set off the spasm of violence was a lie about the deadly knife attack that was promoted again and again online.

The teenager accused of killing three young girls at a Taylor Swift-themed dance class was not — as online agitators claimed — a migrant straight off one of the small boats that bring impoverished people across the English Channel to Britain’s shores. The suspect was born in Wales, to parents who the BBC says came from Rwanda, and the police have not disclosed a motive.

“They are using a tragedy to promote this hate,” said Jasmine Galanakis, 27, who put her young daughter to bed in their home up the street and then joined the crowd on Wednesday evening. “So many people in this community come from different backgrounds, and it’s ignorance driving this. It’s just an excuse for hate, and we won’t stand for it.”

Liverpool, in England’s north, has long been a stronghold of the Labour Party and has a proud working-class tradition. The city’s dock workers have a history of organized action, and particularly after World War II, diversity flourished, making the city among the country’s most multicultural.

The threats in this sliver of Liverpool had been made against Asylum Link Merseyside, the charity that Mr. Roberts manages. He and the staff decided to shut its doors temporarily at the start of the week and bring in carpenters to board up the windows and doors to minimize damage if the building was attacked.

As he watched people gather peacefully in the streets, he said he was moved by the diversity of those who came out to express their support for asylum seekers.

It was especially affirming after years of railing by the former Conservative government against the number of asylum seekers — and its attempt to deport them to Rwanda despite a Supreme Court ruling that the policy was illegal.

Nazehar Benamar, 42, and her cousin Wafa Hizam, 22, who grew up in Liverpool, both said they felt it was important to be there. But they also said they were angry about the violence that erupted in the city center a few days earlier.

“Liverpool is a very multicultural city, but as a person of color, you are always aware of racism and prejudice,” said Ms. Benamar, who is Muslim and wears a hijab. She recalled how as the only nonwhite child in her class, she had been subjected to racial slurs. She said she was saddened that racism and Islamophobia were still so potent so many years later.

“People are being terrorized by fear about this violence,” she said. “Today especially, I could feel it.”

Still, on Wednesday night she was reassured to see members of her local mosque standing alongside university students and retirees. The people of Liverpool had come together to show “what we are made of here,” she said.

What united many of them was the feeling that working-class people are in life’s struggles together. As the evening light turned golden and night slowly set in, one young woman raised a sign that read, “The Enemy of the Working Class Travels By Private Jet Not Migrant Dinghy,” to applause from many standing nearby.

Matty Delaney, 33, who lives just outside Liverpool, said he had heard on Instagram about the demonstration against racism and thought it was important to deliver a clear message to those who had rioted, particularly as a young, white, working-class man.

“We’ve got more in common with an Indian nurse, with a Black bricklayer than we do with the Elon Musks, the Nigel Farages, the Tommy Robinsons, of the world — all these people who are stoking violence,” Mr. Delaney said.

Mr. Musk, the billionaire owner of the social media platform X — where disinformation about the initial attack had been allowed to swirl — threw himself into the fray this week by saying, “Civil war is inevitable” and accusing the prime minister, Keir Starmer, of not protecting “all communities” in Britain.

Mr. Farage, the leader of the populist anti-immigration Reform U.K. party, initially stoked conspiracy theories that drove the riots, before coming out against the violence. And Mr. Robinson, an anti-Islam agitator who founded the English Defense League — originally a street movement, which now spreads Islamophobic and xenophobic views mostly online — was among the far-right figures who pushed for their supporters to take to the streets after the stabbing attack.

By Thursday morning, the rhythm of daily life had returned to Overbury Street. At St. Anne’s Church, next door to the charity for asylum seekers, a local family gathered for a funeral. Discarded placards from the night before lay on the ground nearby.

The staff of the charity was also regrouping, and Mr. Roberts said they were trying to figure out when to reopen. While he said he felt an overwhelming sense of relief that the center had not faced violence, it was difficult to know what would come next.

Speaking of the rioters, he said, “They are trying to damage trust between the community and new arrivals, more than the buildings or infrastructure.” But, he added, “What last night told me was we are a greater value in the community, more than we actually understood, and it was wonderful to see that.”

For now, his staff planned to send a letter of thanks to the community. But they also planned to reinforce the wooden boards that protect the center’s windows, just in case.

The authorities in Vienna have arrested a third teenager in connection with a foiled terrorist attack on a Taylor Swift concert in the city this week. They say they believe that the man, an 18-year-old connected to the main suspect, was not part of the plan but had been in touch with the plotters and had recently sworn allegiance to the Islamic State.

Ms. Swift was scheduled to stage three concerts in Vienna from Thursday through Saturday, but all three performances were canceled after the authorities arrested two teenagers over a plan to attack the sold-out, 50,000-seat stadium. Chancellor Karl Nehammer of Austria said the plot had been designed to leave a “trail of blood.”

Since arresting two other teenagers on Wednesday, the authorities have been racing to investigate the planned attack, although after what the police said was a full confession by the main suspect, they said there was no longer an imminent danger.

The police are looking into a network of people around the main suspect, a 19-year-old Austrian citizen of North Macedonian descent who they said had radicalized himself online and sworn allegiance to the Islamic State. Citing privacy rules, the authorities have declined to name the suspects publicly, but they said that both teenagers arrested on Wednesday were born in Austria and held Austrian citizenship.

During a raid on the main suspect’s house on Wednesday, the police said, officers found chemicals used to make bombs, as well as explosives, timers, machetes, knives and a functioning police siren, which investigators believe he planned to use to gain access to or move around the area around the stadium.

The concert cancellations affected about 200,000 Taylor Swift fans, some of whom had traveled to Europe from other continents to see her perform as part of her Eras Tour. Ms. Swift has not commented publicly on the cancellations.

A 15-year-old boy who was held for questioning on Wednesday about the plot has been released and is being treated as a witness, the police said. They said that they had determined he was not part of the plot but that he knew many of its details and had helped corroborate some key elements of the main suspect’s confession.

Just as she was boarding her flight at Boston Logan International Airport headed for a Taylor Swift concert in Vienna, Mary DePetris excitedly checked the online fan group, Swiftie Nation.

Austrian authorities had discovered a terrorist plot targeting Taylor Swift’s Eras Tour in the city, she read. On Wednesday, just before takeoff, organizers canceled all three shows. Ms. DePetris, 47, stepped onto the plane and broke the news to some of her fellow passengers.

“Half the plane was crying,” Ms. DePetris said. “It’s not just about the shows, it’s the community coming together and feeling safe at her concerts, and Swifties letting their guard down. And this just shifted all of that,” she said. “How can we do that now that we feel we are targeted?”

As the estimated 200,000 people who had been expected to worship at Ms. Swift’s proscenium in Vienna grappled with crushing disappointment, wasted money and a measure of fear at narrowly avoiding danger, a sea of fans flooded the baroque city looking for ways to shake it off.

They traded Eras merchandise in the shadow of the vacant stadium, or dissolved into tears when they caught the strains of Ms. Swift’s stanzas drifting from the doorways of sympathetic gift shops or churches. Some hung handmade friendship bracelets — a treasured Swiftie talisman inspired by a song lyric — on a tree on Corneliusgasse, a central Vienna thoroughfare whose name echoes the title of Ms. Swift’s song “Cornelia Street.” There, hundreds hugged, cried and commiserated in the middle of the road.

Tempering the dejection for many was a feeling that a missed concert was far from the worst outcome possible. On Thursday, Austrian authorities released information on the two teenagers they say planned to attack, outlining a picture of a terrorist assault designed to kill as many people as possible with machetes and explosives, plotted by the pair who had become radicalized by Islamic extremism on the internet.

One had recently started a job for a events service provider that was working at the Ernst Happel Stadium, where Ms. Swift was scheduled to play, according to Franz Ruf, a senior Austrian security official. The suspect, who the authorities did not name but said was 17 years old, was arrested there on Wednesday.

“I feel grateful to be alive,” said Charlotte Keller, 34, a human resources manager from Rome outside the stadium on Thursday.

Ewald Tatar, a manager at Barracuda Music, which organized the Austrian leg of the Eras Tour, said in a news conference that the decision to cancel the concerts was made together with Ms. Swift’s management, based on the information received from authorities.

“Although it was not an everyday decision, it was definitely the right one,” Mr. Tatar said, citing the fact that one of the suspects was an arena employee as a deciding factor. According to Barracuda’s website, all tickets would be automatically refunded within the next two weeks.

In an essay for Elle magazine from 2019, Ms. Swift said her “biggest fear” was the potential for an attack on one of her concerts. “After the Manchester Arena bombing and the Vegas concert shooting, I was completely terrified to go on tour,” she wrote, referring to terrorist attacks at concerts in those cities in 2017 that killed a total of 82 people and wounded hundreds of others.

Ms. Swift wrote at the time that she worried about keeping “3 million fans safe over seven months,” during her Reputation Tour. Her Eras tour will be three times as long, with more than 150 shows over two years that one company estimated could generate $4.6 billion in North America alone.

Ms. Swift has not yet commented publicly on the situation.

Inside the 16th-century Lutheran City Church in Vienna on Thursday, a group of teenagers from the Czech Republic sat in a pew, inconsolable. A sign outside said, “Dear Swifties, we sympathize,” and Ms. Swift’s song “August” blasted through the sanctuary: “I can see us lost in the memory,” she sang. “August slipped away into a moment in time …”

The girls sang, too. And they sobbed.

“We came here carefree,” said one, Katherine Penkavova, 18. “And now we face danger,” she added. The girls leaned their heads down on the back of the pew. “At least we can sing our feelings here,” Ms. Penkavova said.

The city tried its best to dry the floods of tears.

Listings of palliative events popped up almost immediately. Vienna’s Albertina museum and municipal pools offered Swifties free entry, while Austria’s national railway offered refunds on unused train tickets. A dance party called “Shake It Off” invited fans to come dressed in their sparkly concert best. A restaurant offered free flutes of pink sparkling wine for every crushed concertgoer.

For some the concerts had meant more than a silly, good time, including Eliya Briand, 22, and her sister Naomi, 24, who arrived in Vienna on Thursday from Netanya, just north of Tel Aviv, seeking a reprieve from Israel’s war in Gaza. Now thousands of miles from home, the sisters felt they were facing the same fear.

“It has been a really, really difficult year, and this concert was sort of an escape from the reality at home,” Eliya said.

Her sister Naomi said they had come “from war, from terror — and now we meet it again.” She added, “For this concert to be canceled because of that specific reason, it hurts a lot more.”

Some, like Teng Yilin, 22, were making the most of the situation — while blinking back the occasional tear. Ms. Yilin flew in for a single day from Shanghai to live her dream of seeing Ms. Swift live. She arrived before dawn on Thursday and was scheduled to leave around midnight. She got the news about the cancellation on the plane, but didn’t believe it until she saw people crying when she landed.

Wandering Vienna lost in grief before dawn, Ms. Yilin and her boyfriend were taken in by a group of Swifties, some of whom had come from as far away as South Africa. They bought her beers, she said, and the bar owner played Taylor Swift songs.

“At the beginning, it was sad, but after a few hours we were laughing,” Ms. Yilin said. “I’m heartbroken,” she added. “But I think it was still a good night.”

It was still too early to assess the economic fallout from the canceled concerts. The Austrian Hotel Association offered its members legal guidelines to manage an expected influx of cancellations, but Oliver Schenk, a spokesman for the organization, said that he had received conflicting reports.

“It is not yet possible to say how high the financial loss is for the companies,” he said.

Outside the stadium on Thursday, there was no coveted Eras merchandise for sale. Vendors began to pack up tubs of uneaten wurst and untapped kegs of Austrian lager.

Stefan Schneider, 48, the owner of Arena Cocktail Catering, said he had spent 10,000 euros, about $11,000, on hotel rooms for 60 staff members he brought in from Germany for the three-day event, plus another 10,000 euros in cocktail ingredients. The event would have accounted for 30 percent of his yearly income, if all had gone well, he said. He added that he had no insurance.

“It’s a disaster,” Mr. Schneider said, but looming larger than that was his fear that other concerts could be under threat. “It’s a problem. You have thoughts, what about the next event? What about disaster after disaster?”

Next week, the singer’s global tour is scheduled to begin a run of five sold-out shows at Wembley Stadium, a 90,000-seat arena in London. A spokesman for London’s Metropolitan Police said in a statement that there was “nothing to indicate that the matters being investigated by the Austrian authorities will have an impact on upcoming events here in London.”

With all the tickets for the London gigs snapped up and further tour dates fixed through December, it was unlikely that the disappointed Swifties in Vienna would get to see the singer soon.

And without a concert to prep for, they flooded the city.

The Spanish Riding School tours appeared sold out on Thursday, and braceleted people queued for tickets to Schönbrunn Palace. Mozart and Taylor Swift songs competed for earspace across its winding streets. Alex Januschke, a waiter at Cafe Tirolerhof, said he had spent the afternoon managing tables of disconsolate fans.

“My advice?” he said. “See the city!”

Melissa Eddy contributed reporting from Berlin, Christopher F. Schuetze from Leipzig, Germany, and Alex Marshall from London.

When Vice President Kamala Harris’s husband, Doug Emhoff, landed in Paris this week, the Summer Olympics were naturally on the schedule. He met with the champion sprinter Noah Lyles, watched the U.S. men’s basketball team rally against Serbia and is to lead the American presidential delegation at the closing ceremony on Sunday.

But Mr. Emhoff, the first Jewish spouse of an American vice president or president, also used his trip to focus on an issue that is far more sobering and also deeply personal: a surge of antisemitism in the United States and around the world since the war between Israel and Hamas erupted in October.

“It is a poison coursing through the veins of democracy and democratic ideals,” Mr. Emhoff said on Friday at a commemoration of a deadly 1982 attack on a storied Jewish deli in Paris. Six people were killed in that attack, including two Americans, and 22 were wounded.

“Part of fighting hate is living openly and proudly as a Jew and celebrating our faith and our culture,” Mr. Emhoff said as helicopters buzzed overhead, a reminder of the tight security that France has imposed during the Olympics. “I love being Jewish, and I love the joy that comes with being Jewish. And I’m not going to let anyone tell me how to be Jewish.”

Mr. Emhoff has emerged as the Biden administration’s most visible face in the struggle against antisemitism. He has convened Jewish leaders at the White House, called former President Donald J. Trump “a known antisemite,” and drawn on his faith to offer comfort to Jewish Americans anguished by the Oct. 7 Hamas-led attack on Israel.

He is also, of course, the spouse of a presidential candidate. And to that end, he attended a private fund-raiser in Paris for Ms. Harris’s campaign, though he made no direct mention of her presidential bid in his public statements — even to the scores of journalists who watched him grab lunch at L’As du Falafel, a celebrated falafel spot, on Friday.

Mr. Emhoff said only that Ms. Harris had encouraged him to embrace his role as one of the Biden administration’s main advocates for Jewish people. “She and I knew that somebody had to speak out, and she knew that I had to take on this fight, no matter how difficult,” he said.

A day earlier, Mr. Emhoff attended a brief discussion on antisemitism and Holocaust education at the headquarters of UNESCO, the United Nations cultural organization, where he said the Biden administration was working with Congress on a $2.2 million grant for the agency’s program to advance teaching about the Holocaust and genocide. The money will fund training for educators on how to prevent antisemitism and how to react when it occurs in schools, UNESCO said.

During that event, Mr. Emhoff recalled visiting Poland in 2023 and learning about his family history. His great-great-grandparents fled antisemitic persecution, he said, while other family members were killed.

“This is not just some concept in the history books,” he said.

Mr. Emhoff’s visit took place just over a year after the United States rejoined UNESCO, which it had stopped funding in 2011 under the Obama administration after the agency voted to include Palestine as a full member, and then withdrew from completely in 2017 under Mr. Trump.

Audrey Azoulay, UNESCO’s director general, said the United States’ decision to rejoin had been key in supporting programs to stop antisemitism, racism and hate speech. “We now have this opportunity to further strengthen our actions in this regard, something which would have been impossible without your return,” she said.

Given the heightened emotions around the war in Gaza, Ms. Harris has walked a fine line on Israeli and Palestinian issues. When Prime Minister Benjamin Netanyahu of Israel visited Washington last month, Ms. Harris condemned pro-Hamas demonstrators as “despicable” and voiced support for Israel’s right to defend itself against terrorism. But she also said she would not be silent about the suffering and destruction caused by the Israeli military’s campaign in Gaza.

At the commemoration for the 1982 attack, Mr. Emhoff and French officials lit candles and laid wreaths for victims of the lunchtime assault in a crowded deli on the Rue des Rosiers, a narrow, lively street in the heart of the French capital’s old Jewish quarter. The four attackers threw a grenade into the restaurant — which is now a clothing store — and fired at customers before fleeing.

No group ever formally claimed responsibility for the attack, and for decades the case remained unsolved. Then, in 2015, the French authorities said they had identified three suspects, all of whom lived abroad and were accused of being tied to a small Palestinian terrorist organization that had been blamed for deadly attacks in several countries.

One of the suspects, Walid Abdulrahman Abou Zayed, a Norwegian citizen of Palestinian origin, was extradited to France and has been held there since 2020. In February, the Paris Appeals Court rejected an appeal by his lawyers to have the charges against him dropped, but it is unclear when a trial might take place.

If the Olympic Games have made of Paris a midsummer night’s dream, perhaps the Pont du Carrousel has been its heart, a dimly lit bridge over glittering water, a merry-go-round of incarnations as the weeks have passed.

The broad bridge spans the center of Paris, leading from the Quai Voltaire on the Left Bank of the Seine River to three vaulted openings into the Louvre courtyard on the Right Bank. It has always been a place for lovers to linger, joggers to pause, selfie seekers to snap and Paris wanderers to succumb to wonderment.

There are few better places to drink in the city. The Grand Palais and Eiffel Tower rise to the west. To the east loom the domed Académie Française and, in the distance, the Notre-Dame Cathedral, now almost restored after the 2019 fire. The cleaned-up river is ever-changing, now churning after a downpour, now glassy still.

France was in a somber mood through much of the summer. Then the Paris Olympics began two weeks ago, replacing social fracture with patriotic rapture, dissolving fences of division into bridges of understanding, none more unifying than the Pont du Carrousel, at least for now.

From the bridge, crowds gaze nightly at the airborne Olympic cauldron. It is suspended from a hot-air balloon, to form a golden orb trailing misty swirls above the Tuileries Garden in front of the Louvre in what looks like a visitation from some benign extraterrestrial power. Seen from the bridge, the balloon hovers tantalizingly over one wing of the Louvre as if to announce a season of magic.

“What a joy!” exclaimed Thomas Bordeaux, a publishing executive, as he took photographs from the bridge the other night. “I have never seen my city so beautiful.” He had fled Paris at the start of the Games, like many Parisians fearing crowds or even terrorist violence, but returned this week after concluding he had made the wrong call.

Living close to the bridge, I have observed the joyous crowds on it and wondered if they know its murderous past. I will get to that.

The bridge was inaugurated by King Louis-Philippe in 1834 as the Pont des Saints-Pères, only to be blown up a century later to make way for the current bridge, built a little farther west to align with the Louvre entryway. It has always adapted to change, but rarely at the pace of recent weeks.

In the 10 days before the Games, a stand spanning the bridge was erected for the elaborate July 26 opening ceremony. Workers toiled long hours to build the bleachers that disfigured the bridge but offered one of the best views of the flotilla carrying thousands of drenched athletes.

No sooner was that over than the stand was dismantled by crews working through the night. Kristen Faulkner of the United States swept across the bridge last Sunday in a culminating moment of the women’s cycling road race to take the gold medal. Rapturous crowds cheered her and the other leaders of the race as they emerged onto the riverfront from the Place du Carrousel, the square between the wings of the Louvre Palace that gave the bridge its name.

On Saturday, Olympic marathon runners will cross the Place du Carrousel, so named in 1662 after Louis XIV hosted festivities there to mark the birth of his son, before turning down the Right Bank at the Pont du Carrousel, without traversing it, as they head west toward Versailles.

To their left as they turn, the athletes might notice a small bronze plaque attached to the low stone wall at the entry to the bridge. It reads, “To the memory of Brahim Bouraam, 1965-1995, victim of racism, assassinated here on May 1 1995.”

May 1 is a national holiday in France, the country’s Labor Day. Supporters of the extreme-right, anti-immigrant National Front (now the National Rally) had gathered in 1995 at the nearby golden statue of Joan of Arc, the martyred 15th-century military leader. She has been used by the party as a symbol of the battle to keep France French, free of immigrants.

From the statue, they marched to the Pont du Carrousel. Mr. Bouraam, a 29-year-old Moroccan immigrant who worked in a grocery store, had decided to enjoy his holiday strolling along the Seine to meet a friend at the bridge. The meeting never happened. A group of National Front extremists confronted him, hurling abuse. He was shoved into the river and drowned.

President François Mitterrand, a few weeks before the end of his 14 years in office and a few months before his death from cancer, came to the spot to express his sorrow. He tossed a bouquet of lily of the valley into the Seine.

Afterward, for many years, the episode was largely forgotten. Jean-Marie Le Pen’s hate-filled National Front began its tortuous journey toward his daughter Marine Le Pen’s National Rally of today, which vows, and vows again, that racism is gone from its agenda.

I met Said Bouraam, 39, the thoughtful son of the murdered man, this week at the bridge. He pointed to the plaque and said, “For me, this is all that is left of my father.”

He was 9, and living in Morocco with his mother, when a family friend delivered the news. He hardly knew his father but wanted to follow in his footsteps, so he came to France in 2007, later becoming a citizen. “Commemorating my father is important,” he said. “It’s the only way to fight forgetfulness.”

The Paris mayor, Anne Hidalgo, whose fervent belief in the Olympics as a means to uplift Paris and unite people has been vindicated, now conducts an annual commemorative service at the site.

Mr. Bouraam works for the Paris police and is “proud of keeping people safe and this Olympic fête happy for everyone.” He does not believe that the ideas of the National Rally have changed. But, he told me, “I feel no hatred, no desire for vengeance.”

One of the assailants, Mickaël Freminet, served an eight-year prison sentence; three others received lighter prison terms.

So in these bright and beautiful Olympic days I have two images of the Pont du Carrousel and two images of France: the joyous capacity to come together that has made these Olympics so remarkable and, lurking still, the violent division that made the weeks before the Games so difficult — and which could quickly return.

“We are a country that likes to flagellate itself,” Gabriel Attal, the departing prime minister, told me in an interview. “When we compare ourselves with others, it’s more often to despair than to console ourselves. But we are also a country that likes to be proud of itself, and we see that now.”

France has shown the world what it can be; pride and patriotism are rekindled and Paris has cast its spell once more. “The atmosphere is so great, really nothing to do with regular Paris life!” said Lucie Breillat, 26, who works for a cosmetics company, as she stood on the bridge.

There is an old carousel in the Tuileries, not far from where the Olympic cauldron and balloon are during the day. Once, in the midst of the Covid-19 epidemic, I went there.

The park, like Paris, was deserted, but the carousel still turned. Round and round went colorful horses, an ostrich, a car, a plane, a ship and a couple of Cinderella carriages. My partner and I chose horses. The music was North African. There were a couple of children.

The carousel seemed a little miracle in 2021. Paris would be back, I thought then, never imagining how transformative the Olympics might be, nor the degree to which Paris survives and incarnates tolerance precisely as a result of bitter and violent experience.

The Pont du Carrousel, viewed from all its angles, is that Paris, whose hold on the human imagination explains why “We’ll always have Paris” is the most famous line in the movies.

Carles Puigdemont, the fugitive Catalonia pro-independence leader, has dodged the Spanish authorities and fled the country again after popping up in Barcelona this week and holding a public rally, his lawyer said on Friday.

Mr. Puigdemont, who was president of the Catalan government when the region held a 2017 illegal independence referendum, has been living in self-imposed exile since then. Despite an arrest warrant by the Spanish authorities, he returned to Barcelona this week and held a rally near Catalonia’s regional Parliament on Thursday.

The Spanish police launched a manhunt to find him, with roadblocks and checks on cars entering and leaving Barcelona. But on Friday, his lawyer, Gonzalo Boye, said on Catalan radio that Mr. Puigdemont had left Spain.

Mr. Puigdemont had promised to be in Barcelona on Thursday when the regional Parliament’s new president, a Socialist who does not support Catalan independence, was scheduled to be voted in. Experts saw his return as a last-ditch attempt to remain relevant in Catalonia, where for the first time in decades, pro-independence parties did not win an outright majority in regional elections this year.

Jordi Turull, the general secretary of Mr. Puigdemont’s party, said that the leader was headed back to Waterloo, in Belgium, where he has spent at least part of his self-imposed exile. Mr. Turull said that Mr. Puigdemont had been in Spain for a few days before the rally.

The appearance — and mysterious disappearance — of Mr. Puigdemont, a highly polarizing figure, puzzled the country, amusing some and angering others. Many asked how the Spanish authorities could allow him to come back, speak to a crowd of supporters, and then escape without being caught.

Two members of Catalonia’s police force were arrested on suspicion of helping Mr. Puigdemont escape, and the opposition accused Prime Minister Pedro Sánchez of “humiliating” Spain in an episode that highlighted severe security failures.

Mr. Puigdemont’s supporters, though, expressed pride in their leader.

“I am extremely happy that he came, that he managed to talk and that he was able to say what he wanted to say,” said Anna Navarro, a regional lawmaker with Mr. Puigdemont’s party. “It was an incredible day for the independence movement.”