09-03-2026, 11:50 PM
First Steps in Differential Geometry: Riemannian, Contact, Symplectic
Author: Andrew McInerney
Publication date: 2013
Publisher: Springer New York
First Steps in Differential Geometry is an undergraduate introduction to modern differential geometry that treats geometry largely as the study of structures placed on tangent spaces. McInerney begins by building the necessary foundations from linear algebra and multivariable calculus, then develops differential forms and tensors before turning to three major geometric structures: Riemannian, contact, and symplectic geometry. This is somewhat unusual for an introductory text, since many undergraduate books concentrate almost entirely on curves, surfaces, and Riemannian geometry.
The progression is deliberately gradual. The opening chapters review vector spaces, linear transformations, derivatives, manifolds, tangent vectors, differential forms, and tensorial constructions. Riemannian geometry then introduces metrics and the geometric concepts that arise from them, while the final sections expose students to contact and symplectic geometry—subjects that normally appear considerably later in a mathematics curriculum. The book emphasizes understanding what follows from definitions, using concrete examples and constructions alongside proofs. Its intended audience is students who have completed roughly two years of university mathematics, particularly calculus, linear algebra, and differential equations.
A major strength is therefore its role as a bridge between computational undergraduate mathematics and abstract modern geometry. Rather than requiring a large amount of topology and manifold theory beforehand, McInerney develops many of the tools as they become necessary. Reviews cited by Springer particularly praise the clarity of the presentation, illustrative examples, exercises, motivation, and the unusually broad exposure to modern geometric ideas. For a student planning to continue into differential topology, geometric analysis, mathematical physics, Hamiltonian mechanics, or advanced geometry, it provides a strong conceptual foundation.
Main topics
Key takeaways
BOOK
Author: Andrew McInerney
Publication date: 2013
Publisher: Springer New York
First Steps in Differential Geometry is an undergraduate introduction to modern differential geometry that treats geometry largely as the study of structures placed on tangent spaces. McInerney begins by building the necessary foundations from linear algebra and multivariable calculus, then develops differential forms and tensors before turning to three major geometric structures: Riemannian, contact, and symplectic geometry. This is somewhat unusual for an introductory text, since many undergraduate books concentrate almost entirely on curves, surfaces, and Riemannian geometry.
The progression is deliberately gradual. The opening chapters review vector spaces, linear transformations, derivatives, manifolds, tangent vectors, differential forms, and tensorial constructions. Riemannian geometry then introduces metrics and the geometric concepts that arise from them, while the final sections expose students to contact and symplectic geometry—subjects that normally appear considerably later in a mathematics curriculum. The book emphasizes understanding what follows from definitions, using concrete examples and constructions alongside proofs. Its intended audience is students who have completed roughly two years of university mathematics, particularly calculus, linear algebra, and differential equations.
A major strength is therefore its role as a bridge between computational undergraduate mathematics and abstract modern geometry. Rather than requiring a large amount of topology and manifold theory beforehand, McInerney develops many of the tools as they become necessary. Reviews cited by Springer particularly praise the clarity of the presentation, illustrative examples, exercises, motivation, and the unusually broad exposure to modern geometric ideas. For a student planning to continue into differential topology, geometric analysis, mathematical physics, Hamiltonian mechanics, or advanced geometry, it provides a strong conceptual foundation.
Main topics
- Linear algebra and advanced multivariable calculus
- Tangent spaces and smooth geometric structures
- Differential forms and tensors
- Riemannian geometry
- Contact geometry
- Symplectic geometry
Key takeaways
- Modern approach: treats differential geometry through structures on tangent spaces rather than focusing only on classical curves and surfaces.
- Unusually broad: introduces Riemannian, contact, and symplectic geometry in a single undergraduate text.
- Accessible prerequisites: mainly calculus, linear algebra, and differential equations.
- Good preparation for advanced mathematics: particularly manifolds, differential topology, geometric analysis, and mathematical physics.
BOOK
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