This paper studies two isometric problems between unit spheres of Banach spaces. In the first part, we introduce and study the Figiel type problem of isometric embeddings between unit spheres. However, the classical Figiel theorem on the whole space cannot be trivially generalized to this case, and this is pointed out by a counterexample. After establishing this, we find a natural necessary condition required by the existence of the Figiel operator. Furthermore, we prove that when $X$ is a space with the T-property, this condition is also sufficient for an isometric embedding $T: S_X\rightarrow S_Y$ to admit the Figiel operator. This answers the Figiel type problem on unit spheres for a large class of spaces. In the second part, we consider the extension of bijective $\varepsilon$-isometries between unit spheres of two Banach spaces. It is shown that every bijective $\varepsilon$-isometry between unit spheres of a local GL-space and another Banach space can be extended to be a bijective $5\varepsilon$-isometry between the corresponding unit balls. In particular, when $\varepsilon=0$, this recovers the MUP for local GL-spaces obtained in [40].
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