Differential & Riemannian Geometry

Prerequisite path Preparation & skip guidance · Linear Algebra · Calculus · Point-Set Topology

Studying Differential & Riemannian Geometry

Classical differential geometry relates derivatives to the shape of curves and surfaces. Shifrin develops arclength, moving frames, fundamental forms and geodesics with calculations and selected assistance for exercises. Petrunin and Zamora Barrera take a visual, problem-centered route through the same classical setting. Choose according to whether your main difficulty is carrying out the local calculation or understanding the geometric claim it expresses.

What to work on

Reparametrize a regular curve by arclength and compute curvature and torsion.
Compute first and second fundamental forms of a regular surface.
Relate principal curvatures to Gaussian and mean curvature.
Distinguish intrinsic surface geometry from features of an embedding.

A starting text

Differential Geometry: A First Course in Curves and Surfaces — Theodore Shifrin. Read chapter 1, sections 1-2 on arclength and Frenet frames; chapter 2, sections 1-2 on the fundamental forms and Gauss map; then chapter 2, section 4 on covariant differentiation and geodesics. Chapter 3, section 1 introduces holonomy and Gauss-Bonnet.

Choosing what to read

Parametrize a helix by arclength and compute its curvature before reading the general frame formulas. On a sphere and a cylinder, calculate the fundamental forms and Gaussian curvature, tracking the chosen normal and shape-operator convention. Compare a curve's ambient acceleration with its covariant derivative on the surface. Draw the geometry, then annotate the drawing with the quantities you computed. Plan a separate smooth-manifold and tensor course if the next goal requires coordinate-independent constructions in higher dimensions.

Differential Geometry: A First Course in Curves and Surfaces

Theodore Shifrin

Relate arclength, fundamental forms and geodesics to explicit examples.

How it reads: Calculational classical course with moving frames and surface geometry.
What it expects: Multivariable calculus and linear algebra.
What to keep in mind: Curves and surfaces do not replace a full smooth-manifold and tensor course; signed quantities depend on conventions.
Practice: Public exercises and selected answers, hints or solutions are included. Asterisked exercises are the ones marked for back-of-book assistance; a complete worked solution key is not claimed. Exercises in 1.1-1.2 and 2.1-2.2; consult Answers to Selected Exercises at printed page 121 only for marked questions.

Complete public PDF.

Edition, reading range and source

Read chapter 1, sections 1-2 on arclength and Frenet frames; chapter 2, sections 1-2 on the fundamental forms and Gauss map; then chapter 2, section 4 on covariant differentiation and geodesics. Chapter 3, section 1 introduces holonomy and Gauss-Bonnet.

Source · Checked 2026-10-16

What Is Differential Geometry? Curves and Surfaces

Anton Petrunin and Sergio Zamora Barrera

Investigate the geometric content of a curve or surface calculation.

How it reads: Visual, problem-centered classical geometry emphasizing essential techniques.
What it expects: Multivariable calculus and linear algebra.
What to keep in mind: The authors describe the presentation as elementary and virtually rigorous; use a fuller argument when a proof detail matters.
Practice: The catalogue describes a problem-centered text; public solution coverage was not recorded.

Complete author lecture text on arXiv, version 7 dated December 31, 2025.

Edition, reading range and source

Select the curve and surface problems corresponding to arclength and curvature; use the current archive version's contents rather than assuming Shifrin's numbering.

Source · Checked 2026-10-17

Practice

Curve and surface exercises with selected answers. Exercises in 1.1-1.2 and 2.1-2.2; consult Answers to Selected Exercises at printed page 121 only for marked questions.. Public exercises and selected answers, hints or solutions are included. Asterisked exercises are the ones marked for back-of-book assistance; a complete worked solution key is not claimed.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

For r(t)=(cos(t),sin(t),t), compute speed, curvature and torsion and give an arclength parametrization.
A hint
Use curvature |r' cross r''|/|r'| cubed and torsion det(r',r'',r''')/|r' cross r''| squared. State the torsion convention.
Work through the solution
For $t\in\mathbb R$ , $r'=(-\sin t,\cos t,1)$ , $r''=(-\cos t,-\sin t,0)$ and $r'''=(\sin t,-\cos t,0)$ . Thus $|r'|=\sqrt2$ , while $r'\times r''=(\sin t,-\cos t,1)$ has norm $\sqrt2$ . The curve is regular with nonzero curvature, and

$\kappa=\frac{|r'\times r''|}{|r'|^3}=\frac12.$

Using the convention $\tau=\det(r',r'',r''')/|r'\times r''|^2$ , the scalar triple product is $\sin^2t+\cos^2t=1$ , giving $\tau=1/2$ . The signed arclength coordinate based at $t=0$ is $s=\int_0^t|r'(u)|\,du=\sqrt2t$ . Hence $\widetilde r(s)=(\cos(s/\sqrt2),\sin(s/\sqrt2),s/\sqrt2)$ , and direct differentiation gives $|\widetilde r'(s)|=1$ .

If this was the sticking point:
- Calculus — Review derivatives of vector-valued curves and arclength.
- Linear Algebra — Review cross products, determinants and Euclidean norms.

Already comfortable with this? Skip curve calculations if arclength and the Frenet frame are reliable. Skip the surface review if you can derive the fundamental forms and explain why a cylinder is locally flat intrinsically despite bending in space.

Review the prerequisites · Compare the full subject collection

Where this can lead

Point-Set Topology — Review charts and underlying spaces before a smooth-manifold course.
Symplectic Geometry — Add smooth manifolds, differential forms and de Rham ideas before symplectic structures.

Reading-source dates

Differential Geometry: A First Course in Curves and Surfaces · 2026-10-16

Reading choices revised 2026-10-17.

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What Is Differential Geometry? Curves and Surfaces

PDF

Anton Petrunin and Sergio Zamora Barrera · Lecture text · Introductory

A visual, problem-centered approach to classical curves and surfaces. Emphasizes essential techniques; the authors describe its presentation as elementary and 'virtually rigorous'.

Preparation: Multivariable calculus; linear algebra.

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Lectures on Spaces of Nonpositive Curvature

PDF

Werner Ballmann; appendix by Misha Brin · Monograph / author text · Advanced

Develops nonpositive curvature, boundaries at infinity and rank rigidity, with an appendix on geodesic-flow ergodicity. A specialized continuation, not a general first Riemannian geometry textbook.

Preparation: Riemannian geometry; topology; some dynamical systems.

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Further print reading (2 retained references)

From the original catalogue. These are bibliographic references, not free full-text links; an authorized complete free edition is not currently linked here.

Differential Geometry of Curves and Surfaces — Manfredo P. do Carmo
Differential Geometry — Paulo Ventura Araújo

Additional reading

Search the complete catalogue.

Introductory reading

Flatland: A Romance of Many Dimensions

Edwin Abbott Abbott · Mathematical fiction · Introductory · English · Online text

Complete illustrated HTML at Project Gutenberg.

A two-dimensional narrator encounters other dimensions in a geometric story with Victorian social satire.

Preparation: Elementary geometry.

Use: Enjoy a historical imaginative introduction to dimension.

Open online text / archive Source / terms

Edition and sources

Original title: Flatland: A Romance of Many Dimensions

Version: 1884 novella; illustrated Project Gutenberg edition 45506 · Checked 2026-10-13

The Fourth Dimension

Charles Howard Hinton · Historical popular mathematics · Introductory · English · Online text

Complete HTML at Project Gutenberg.

Geometric models and visual exercises explore higher-dimensional space alongside early twentieth-century speculation.

Preparation: Elementary geometry; treat physical speculation in historical context.

Use: Practice spatial imagination while reading historical popular mathematics.

Open online text / archive Source / terms

Edition and sources

Version: 1906 second edition; Project Gutenberg transcription · Checked 2026-10-13

Intermediate reading

Differential Geometry: A First Course in Curves and Surfaces

Theodore Shifrin · Textbook · Intermediate · English · PDF

The complete PDF opens on the author’s or university’s public site.

Develops curves, surface curvature, geodesics and Gauss–Bonnet, with exercises and selected solutions. Later sections introduce hyperbolic geometry and variational problems.

Preparation: Multivariable calculus; linear algebra; elementary ODE.

Use: Study before abstract Riemannian geometry.

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Edition and sources

Version: April 2021 · Checked 2026-10-13

Graduate reading

Lectures on Complex Analytic Manifolds

Laurent Schwartz · Long lecture notes · Graduate · English · PDF

Treats analytic manifolds and their local structure. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Complex analysis; topology.

Use: Classical geometric analysis preparation.

Studying Differential & Riemannian Geometry

What to work on

A starting text

Choosing what to read

Practice

Try these before moving on

Where this can lead

Additional reading

Introductory reading

Intermediate reading

Graduate reading

Research reading