0

1

2

n

The set of all tangent vectors to the manifold at that point

The set of all normal vectors to the manifold at that point

The set of all vectors in the manifold at that point

The set of all vectors in the tangent bundle of the manifold at that point

A map that is continuous and differentiable

A map that is continuous and has a continuous derivative

A map that is continuous and has a differentiable derivative

A map that is continuous and has a continuous second derivative

If a smooth map is one-to-one and onto, then it has a smooth inverse

If a smooth map is one-to-one, then it has a smooth inverse

If a smooth map is onto, then it has a smooth inverse

If a smooth map is continuous, then it has a smooth inverse

A vector field whose components are smooth functions

A vector field whose components are continuous functions

A vector field whose components are differentiable functions

A vector field whose components are integrable functions

The set of all integral curves of the vector field

The set of all tangent vectors to the vector field

The set of all normal vectors to the vector field

The set of all vectors in the vector field

The derivative of the vector field along its flow

The derivative of the vector field along its integral curves

The derivative of the vector field along its tangent vectors

The derivative of the vector field along its normal vectors

A smooth section of the tangent bundle

A smooth section of the cotangent bundle

A smooth section of the tensor bundle

A smooth section of the exterior bundle

The derivative of the differential form along its flow

The derivative of the differential form along its integral curves

The derivative of the differential form along its tangent vectors

The derivative of the differential form along its normal vectors

The cohomology of the exterior algebra of the differential forms on the manifold

The cohomology of the tangent bundle of the manifold

The cohomology of the cotangent bundle of the manifold

The cohomology of the tensor bundle of the manifold

A theorem that states that any differential form on a smooth manifold can be decomposed into a sum of exact, closed, and coexact forms

A theorem that states that any differential form on a smooth manifold can be decomposed into a sum of exact and closed forms

A theorem that states that any differential form on a smooth manifold can be decomposed into a sum of exact and coexact forms

A theorem that states that any differential form on a smooth manifold can be decomposed into a sum of closed and coexact forms

A theorem that relates the curvature of a surface to its Euler characteristic

A theorem that relates the curvature of a surface to its genus

A theorem that relates the curvature of a surface to its area

A theorem that relates the curvature of a surface to its volume

A theorem that relates the homology of a smooth manifold to its cohomology

A theorem that relates the homology of a smooth manifold to its de Rham cohomology

A theorem that relates the cohomology of a smooth manifold to its homology

A theorem that relates the cohomology of a smooth manifold to its de Rham cohomology

A conjecture that states that every smooth manifold admits a Morse function

A conjecture that states that every smooth manifold admits a smooth vector field

A conjecture that states that every smooth manifold admits a differential form

A conjecture that states that every smooth manifold admits a Riemannian metric

A conjecture that states that every simply connected, closed, smooth 3-manifold is homeomorphic to the 3-sphere

A conjecture that states that every simply connected, closed, smooth 4-manifold is homeomorphic to the 4-sphere

A conjecture that states that every simply connected, closed, smooth 5-manifold is homeomorphic to the 5-sphere

A conjecture that states that every simply connected, closed, smooth 6-manifold is homeomorphic to the 6-sphere