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Are diffeomorphisms of sphere to itself which preserve area element rotations?

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    I believe this is not the case when the dimension is greater than 1; Think of a smooth diffeomorphism which restricts to rotations on each height level set but not a global rotation. – 2017-02-24
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    What's "additional context"? – 2017-03-02
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    @user53216: "Additional context" generally means some subset of "Where did the question arise?" or "Is the question homework?", "What background do you have?" and "What tools are available?", "What have you tried, and where are you stuck?" These pieces of information help people pitch answers at a suitable level, or with a suitable amount of detail. – 2017-05-05

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No: the group of rotations is finite dimensional, whereas the group of area preserving is not. Here is a concrete example. If $f$ is a smooth function, let $X$ be the unique vector field such that $i(X).\omega =df$, where $\omega$ is the volume form, then the flow of $X$ preserves the volume form (it is Hamiltonian). One can choose $f$ constant in the north hemisphere, not in the south hemisphere, so its flow is the identity in the north hemisphere and not in the south hemisphere.