# Differential geometry vs differential topology

• sabw1992
Riemannian and symplectic geometry- the use of differential equations and calculus to study geometric objectsIn summary, differential topology focuses on the study of smooth manifolds without a metric, while differential geometry studies smooth manifolds with a metric and covers a wide range of topics such as Riemannian and symplectic geometry. Differential topology also looks at the invariants of smooth manifolds under diffeomorphisms, while differential geometry uses differential equations and calculus to study geometric objects.
sabw1992
in a nutshell, what is the difference between those two fields?

I think of differential topology as the study of the properties of smooth manifolds that do not depend upon a metric. One might say that it is the study of invariants of smooth manifolds under diffeomorphsms. Differential geometry on the other hand studies the properties of smooth manifolds that do depend upon a metric, particuarly a smooth metric tensor on a smooth bundle over the manifold.

These two definition are overly restricitve but capture the basic idea of the difference.

Classical subjects in Differential topology are

- What are the cobordism classes of smooth manifolds?

- what are the topological invariants of smooth manifolds?

- when does a combinatorial manifolds admit a smooth structure?

- What are the smooth structures on a manifold?Differential geometry covers a vast array of subjects

## What is the difference between differential geometry and differential topology?

Differential geometry is a branch of mathematics that studies smooth curves and surfaces using concepts from calculus and linear algebra. It focuses on the geometric properties of these objects, such as curvature and length. Differential topology, on the other hand, is a branch of mathematics that studies the topology of smooth manifolds, which are generalizations of curves and surfaces. It focuses on the global properties of these objects, such as connectedness and compactness.

## How are differential geometry and differential topology related?

Differential geometry and differential topology are closely related fields, with differential topology providing the foundational concepts for differential geometry. Differential geometry uses the tools and techniques of differential topology to study the geometric properties of curves and surfaces, while differential topology uses the concepts and results of differential geometry to study the global properties of manifolds.

## What are some applications of differential geometry and differential topology?

Differential geometry has many applications in physics, engineering, and computer graphics, where it is used to model and analyze the behavior of physical systems. Differential topology has applications in topology and geometry, as well as in theoretical physics and cosmology, where it is used to study the structure of the universe.

## What are some key concepts in differential geometry and differential topology?

In differential geometry, some key concepts include curvature, geodesics, and Riemannian metrics. In differential topology, some key concepts include manifolds, homotopy, and cohomology. Both fields also make use of concepts from calculus, linear algebra, and topology.

## What are some famous theorems in differential geometry and differential topology?

In differential geometry, some famous theorems include the Fundamental Theorem of Curves, the Gauss-Bonnet Theorem, and the Nash Embedding Theorem. In differential topology, some famous theorems include the Poincaré Conjecture, the Brouwer Fixed Point Theorem, and the Hairy Ball Theorem. These theorems have important implications in both fields and have influenced other areas of mathematics as well.

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