Differential Geometry: Mapping Surfaces: The Metric (Part 1 of 2)

### How Do You Measure Distance on a Curved Surface?

If a surface is curved—but you’re stuck living on it—how do you even define distance?

We’re studying Visual Differential Geometry and Forms — Tristan Needham, focusing on intuition first, formulas second.

Last meeting: Gaussian curvature and intrinsic geometry—how surfaces detect curvature.

This meeting: the metric—how geometry is encoded in ds², and how maps of the sphere reveal distortion, distance, and curvature.

### What We’ll Cover

• Why maps distort geometry (projective sphere)
• The metric as the fundamental object ds²
• How coordinates encode distance
• First glimpse: curvature from the metric

### What to Expect

• Calculus required (partial derivatives, multivariable thinking)
• Reading Chapter 4.1–4.4 encouraged
• Problems will be worked selectively (focus on key ideas)
• Discussion-based (not a lecture)

### Why It Matters

The metric is the foundation of:
• General relativity (spacetime geometry)
• Gauge theory
• Modern geometry and physics

### Important

Discussion will stay focused on the agenda.
For casual physics conversation, see other group meetings.

### Optional References

Weeks • Baez & Muniain • Greenberg • Isham

### More in Physics With Friends

This event is one of many collaborative study tracks in our Physics With Friends community.
Explore other topics and join additional study groups here:
https://www.meetup.com/physicswithfriends/events/
Join anytime — come prepared to think.