A manifold is a topological space that is locally Euclidean.

What does locally Euclidean mean in the context of manifolds?

Locally Euclidean means that every point in the space has a neighborhood that is homeomorphic to Euclidean space.

What are some examples of manifolds?

Examples of manifolds include spheres, tori, and projective spaces.

What is the dimension of a manifold?

The dimension of a manifold is the number of coordinates needed to describe each point in the space.

What is a smooth manifold?

A smooth manifold is a manifold where the transition functions between coordinate charts are infinitely differentiable.

What is a topological manifold?

A topological manifold is a manifold where the transition functions between coordinate charts are continuous.

What is a differentiable manifold?

A differentiable manifold is a manifold where every point has a neighborhood that is diffeomorphic to an open subset of Euclidean space.

What is a tangent vector on a manifold?

A tangent vector on a manifold is a derivation of the algebra of differentiable functions at a point.

What is a Riemannian manifold?

A Riemannian manifold is a manifold equipped with a positive definite metric tensor.

What is a symplectic manifold?

A symplectic manifold is a manifold equipped with a closed nondegenerate 2-form.

What is a complex manifold?

A complex manifold is a manifold that is locally biholomorphic to an open subset of complex Euclidean space.

What is the Poincaré conjecture?

The Poincaré conjecture is a conjecture in topology that every simply connected, closed 3-dimensional manifold is homeomorphic to the 3-sphere.

What is the classification of manifolds in low dimensions?

In dimension 1, all manifolds are classified as circles. In dimension 2, all manifolds are classified as spheres, projective planes, or tori. In dimension 3, manifolds are much more complicated and there is no complete classification.

What is a closed manifold?

A closed manifold is a manifold without boundary.

What is an open manifold?

An open manifold is a manifold that is not compact.

What is a manifold in mathematics?

A manifold is a topological space that is locally Euclidean.

What does locally Euclidean mean in the context of manifolds?

Locally Euclidean means that every point in the space has a neighborhood that is homeomorphic to Euclidean space.

What are some examples of manifolds?

Examples of manifolds include spheres, tori, and projective spaces.

What is the dimension of a manifold?

The dimension of a manifold is the number of coordinates needed to describe each point in the space.

What is a smooth manifold?

A smooth manifold is a manifold where the transition functions between coordinate charts are infinitely differentiable.

What is a topological manifold?

A topological manifold is a manifold where the transition functions between coordinate charts are continuous.

What is a differentiable manifold?

A differentiable manifold is a manifold where every point has a neighborhood that is diffeomorphic to an open subset of Euclidean space.

What is a tangent vector on a manifold?

A tangent vector on a manifold is a derivation of the algebra of differentiable functions at a point.

What is a Riemannian manifold?

A Riemannian manifold is a manifold equipped with a positive definite metric tensor.

What is a symplectic manifold?

A symplectic manifold is a manifold equipped with a closed nondegenerate 2-form.

What is a complex manifold?

A complex manifold is a manifold that is locally biholomorphic to an open subset of complex Euclidean space.

What is the Poincaré conjecture?

The Poincaré conjecture is a conjecture in topology that every simply connected, closed 3-dimensional manifold is homeomorphic to the 3-sphere.

What is the classification of manifolds in low dimensions?

In dimension 1, all manifolds are classified as circles. In dimension 2, all manifolds are classified as spheres, projective planes, or tori. In dimension 3, manifolds are much more complicated and there is no complete classification.

What is a closed manifold?

A closed manifold is a manifold without boundary.

What is an open manifold?

An open manifold is a manifold that is not compact.