Pittsburgh ? At the end of this week, 20 world leaders will convene here for a global summit to take the measure of the international economy and examine what they can do to irrigate the green shoots and fertilize the fallow areas. In a month of big moments for President Barack Obama, this will be one of the most important.

So much is at stake — for the president, for the economy, for the globe, even for Pittsburgh, which is set to strut its stuff at a summit meant to shine a light on technological innovation, economic revitalization and all things green. So as the politicians and correspondents pack their bags, it might be useful to have a look at the landscape before the world leaders arrive with their sherpas, security details and spouses.

The president’s appearance as host of the G-20 will come after this month’s health-care address to a joint session of Congress and speech before the United Nations. In these Washington and New York events, his leadership will have been tested in both political and diplomatic forums. At the G-20 in Pittsburgh the leadership test will be more subtle, a combination of statecraft and stagecraft.

Ordinarily an American president is at the center of such gatherings; he presides over what still is the world’s largest economy, and despite the growth of China, India, Brazil and others, the United States remains the straw that stirs the economic drink. So a large role on his home turf is to be expected from Mr. Obama, who may be the most recognizable and most popular figure in the world.

That is a lot of pressure for a man who will have been president for only nine months. Pittsburgh is a chance for Mr. Obama to assert his leadership.

There are important domestic implications for the president; his approval rating at home has declined across nearly every major demographic and political group, according to the Pew Research Center, and the surprising thing is not the drop among Republicans (12 points) nor even the decline among independents (9 points). The worrisome thing for the White House is the drop among Democrats (10 points). So this is an opportunity.

Mr. Obama began the G-20 process by convening the meeting in an unexpected venue, a place of steel, the Steelers and a great symphony orchestra. But to the poetry of the place it is for the president to add the prose of vision. His remarks the other day, seizing on the symbolism of the summit site, were a start. “In a place known as the city of bridges,” the president said of Pittsburgh, “we can come together to advance our common interest in a global recovery, while turning the page to a truly 21st-century economy.”

There is plenty of reason for optimism on the economy — but just as much reason for fear. For every positive factor (durable goods are way up, for example) there is a negative one (consumer spending still is down, while consumer debt remains high), and so the recovery, if indeed it has arrived, is unusually fragile and tentative. G-20 finance ministers agreed earlier this month to keep their feet on the stimulus pedal — and to pedal away from limits on bankers’ bonuses, a symbol of how difficult it is for 20 different economies to come to agreement even amid economic crisis.

Nevertheless, all eyes will be on Pittsburgh, if only for 36 hours, as presidents, prime ministers and kings seek to assure their jittery publics that the world has stepped away from the economic precipice — not too dramatic a phrase, by the way, to describe how close to catastrophe the world economy came last autumn.

Here again stagecraft might rule supreme. It is great theater to imagine that 20 world leaders will crowd into a meeting room in the David L. Lawrence Convention Center and emerge with some grand formula to cure what ails the world’s economies. It is a great fiction as well. There will be few unexpected moments at this summit, one of the most scripted dramas in Pittsburgh since the last of August Wilson’s plays was performed. So abandon any notion that there might be spontaneous combustion on the banks of the Allegheny River.

Instead, the body language — and the soothing language — of the world leaders is what will command attention. Do they project crisis or confidence? Are they bathed by a sense of urgency or serenity?

Remember how John F. Kennedy was rattled by his meeting with Nikita S. Khrushchev in Vienna in 1961, an event that might conjure an epigram from Shakespeare: “Exit, pursued by a bear.” Nobody wants the world leaders to depart Pittsburgh International with the frown of fear on their faces.

What the world yearns for is a sense of stability and confidence, two elements that have their utility in the emotional world but that also provide the oxygen for the commercial and investment worlds. Nothing moves in a positive direction — not the stock market, not consumer goods, not the overall economy — without those two things. There is no formula for creating them, which is one of the reasons they have been so elusive these past 12 months.

Over the centuries, Pittsburgh has been known for manufacturing. Its coal, coke, iron and steel created a great economic empire, provided the arms that fought every American war (and not a few foreign ones) from 1861 to our own age and helped produce the goods that created a mass consumer market and culture.

Now a new Pittsburgh, no longer a center of heavy industry but one of technology, health care and education, is being called upon to manufacture something quite different, more of an atmosphere than an arsenal. That’s why there remains some drama in the summit on the Allegheny, where so little may happen but where so much is at stake.