The first full day of the cease-fire passed on Tuesday with no notable violations, but an atmosphere of deep mistrust prevailed in Syria’s war-ravaged areas, residents and monitoring groups reported. The skepticism was fed partly by what appeared to be delays in expediting United Nations aid deliveries to the northern city of Aleppo and other combat zones where civilians have long been deprived of food and medicine. Relief for these areas is an important component of the cease-fire agreement.

The United Nations mediator in the conflict, Staffan de Mistura, told reporters in Geneva that there had been a “significant drop in violence” since the cease-fire plan took effect at sundown on Monday. But he said United Nations relief trucks bound for Aleppo, lined up at southern Turkey’s border with Syria, had not received assurances of safe travel. Mr. de Mistura also said the Syrian government had still not provided required authorizations for deliveries to other locations, “but we are eagerly hoping and expecting the government to issue them very soon.”

The divide between Mr. Kerry and Mr. Carter reflects the inherent conflict in Mr. Obama’s Syria policy. The president has come under increased fire politically for his refusal to intervene more forcefully in the five-year civil war, which the United Nations says has killed more than 400,000 people, displaced more than six million and led to a refugee crisis in Europe. But keeping large numbers of American ground forces out of Syria has also created space for Russia to assume a greater role there, both on the battlefield and at the negotiating table.

The result is that at a time when the United States and Russia are at their most combative posture since the end of the Cold War, the American military is suddenly being told that it may, in a week, have to start sharing intelligence with one of its biggest adversaries to jointly target Islamic State and Nusra Front forces in Syria.

“I remain skeptical about anything to do with the Russians,” Gen. Philip M. Breedlove, who recently stepped down as NATO’s supreme allied commander, said Monday in an interview. “There are a lot of concerns about putting out there where our folks are.”

In an email, Peter Cook, the Pentagon press secretary, said: “As Secretary Carter has said, Secretary Kerry has worked tirelessly to try and ease the suffering of the Syrian people and bring about a diplomatic resolution to the Syrian civil war. If fully implemented, this arrangement could advance those important goals.