---
title: Riemannian Metrics
description: Riemannian metrics define inner products on tangent spaces, allowing the measurement of angles, distances, and curvature on manifolds. Learners will understand how to compute arc lengths, volume elements, and curvature tensors.
category: mathematics
subcategory: differential-geometry
difficulty: beginner, intermediate, advanced
url: /subject/riemannian-metrics
---

# Riemannian Metrics

Riemannian metrics define inner products on tangent spaces, allowing the measurement of angles, distances, and curvature on manifolds. Learners will understand how to compute arc lengths, volume elements, and curvature tensors.

## Available Resources

2 Books • 2 Websites

## Websites

### 1. nLab: Riemannian Geometry

nLab research-wiki entry on Riemannian geometry, defining Riemannian metrics and manifolds and linking them to Levi-Civita connections, curvature, pseudo-Riemannian and Cartan geometry. Useful for seeing how classical metric geometry fits within modern higher-level frameworks, with references.

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

**Link:** https://ncatlab.org/nlab/show/Riemannian+geometry

**Tags:** riemannian-geometry, differential-geometry, riemannian-metric, manifolds, category-theory

### 2. 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

## Books

### 1. Riemannian Geometry

**Author:** Manfredo P. do Carmo

Graduate textbook covering Riemannian metrics, affine and Levi-Civita connections, geodesics, curvature, Jacobi fields, the Hopf-Rinow and Hadamard theorems, and comparison theorems. Readers who know basic smooth manifolds learn to compute with curvature and prove classical global results.

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

**Link:** https://www.amazon.com/dp/0817634908?tag=edmonddante07-20

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

### 2. Riemannian Manifolds: An Introduction to Curvature

**Author:** John M. Lee

Graduate text that develops Riemannian metrics, connections, geodesics and curvature with a clear focus on what curvature means geometrically, culminating in Gauss-Bonnet, Cartan-Hadamard and Bonnet-Myers. Readers learn to prove core comparison theorems from first principles.

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

**Link:** https://www.amazon.com/dp/0387983228?tag=edmonddante07-20

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

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