---
title: Surfaces
description: This topic covers two-dimensional topological manifolds and algebraic varieties. Learners will understand their classification, geometric properties, and how to analyze complex structures, algebraic curves, and coordinate charts on these fundamental spaces.
category: mathematics
subcategory: algebraic-geometry
difficulty: beginner, intermediate, advanced
url: /subject/surfaces
---

# Surfaces

This topic covers two-dimensional topological manifolds and algebraic varieties. Learners will understand their classification, geometric properties, and how to analyze complex structures, algebraic curves, and coordinate charts on these fundamental spaces.

## Available Resources

1 Books • 4 Websites

## Websites

### 1. nLab: Algebraic Surface

Overview and references

**Difficulty:** Beginner | **Price:** Free

**Link:** https://ncatlab.org/nlab/show/algebraic+surface

### 2. Stacks Project

Open-source, collaboratively written reference on algebraic geometry and algebraic stacks, with every result tagged and proved in full. This entry is a lemma computing the normal and conormal sheaves of an effective Cartier divisor, a basic tool for studying curves on surfaces.

**Difficulty:** Advanced | **Price:** Free

**Link:** https://stacks.math.columbia.edu/tag/0B3P

**Tags:** algebraic-geometry, cartier-divisors, sheaves, schemes, algebraic-surfaces

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

### 4. nLab

Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics.

**Difficulty:** Advanced | **Language:** English | **Price:** Free

**Link:** https://ncatlab.org

**Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki

## Books

### 1. Differential Geometry of Curves and Surfaces

**Author:** Manfredo P. do Carmo

Classical treatment of curves and surfaces in three-dimensional space: curvature, the Gauss map, intrinsic geometry, geodesics, and the Gauss-Bonnet theorem. Completing it gives the concrete surface intuition that abstract Riemannian geometry and manifold theory later generalize.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0132125897?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, differential-geometry

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