Let $ M $ be a second-countable and Hausdorff complex manifold, and $ N $ be its complex submanifold. That is, we assume that for all $ x \in N $ there exist an open neighbourhood $ U $ of $ x $ in $ M $ and a chart $ \phi\colon U \to \mathbb{C}^n $ such that $ \phi(U \cap N) = \phi(U) \cap \mathbb{C}^k $. As usual, we consider $ N $ as a complex manifold by declaring that all such $ \phi|_{U \cap N} $ are charts on $ N $.
Question 1. Can every holomorphic function $ f\colon O \to \mathbb{C} $ on an open subset $ O $ of $ N $ be extended to an holomorphic function on an open subset of $ M $?
Question 2. How about real-analytic manifolds and real-analytic functions?
This holds for differentiable manifolds: we can extend $ f $ locally and glue them by using a partition of unity. However, we cannot use a partition of unity on complex or real-analytic manifolds.
Background. I am reading R. O. Wells, Jr.’s Differential Analysis on Complex Analysis (3rd edition). In this book, the structure sheaf on a submanifold $ N $ is defined to be the “restriction” of that on $ M $ (Definition 1.3 and the following). I wonder if Wells’ definition is correct.