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Let X be a compact connected Riemann surface of genus g, with g ≥ 2. For each d < η(X), where η(X) is the gonality of X, the symmetric product Sym d (X) embeds into Pic d (X) by sending an effective divisor of degree d to the corresponding holomorphic line bundle. Therefore, the restriction of the flat Kähler metric on Pic d (X) is a Kähler metric on Sym d (X). We investigate this Kähler metric on Sym d (X). In particular, we estimate it's Bergman kernel. We also prove that any holomorphic automorphism of Sym d (X) is an isometry.

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