Differential Geometry Quizzes for Master's Students

Question 1

Define a smooth manifold.

Accepted Answer

A surface where the tangent plane is well-defined at every point

Answer

A collection of points with a continuous distance function

Answer

A collection of curves that are differentiable at every point

Answer

A surface with a constant Gaussian curvature

Question 2

What does the curvature tensor of a Riemannian manifold measure?

Accepted Answer

The deviation of the manifold from flatness

Answer

The direction of curvature

Answer

The total curvature of the manifold

Answer

The second derivatives of the metric tensor

Question 3

What is a consequence of the Gauss-Bonnet theorem?

Accepted Answer

The total curvature of a compact surface is a topological invariant

Answer

The Gaussian curvature of a surface is constant

Answer

The geodesic curvature of a curve on a surface is zero

Answer

The Gauss map of a surface is a diffeomorphism

Question 4

What differential form measures the volume of a Riemannian manifold?

Accepted Answer

Volume form

Answer

Curvature form

Answer

Symplectic form

Answer

Kähler form

Question 5

Which theorem relates the curvature of a Riemannian manifold to its topology?

Accepted Answer

Gauss-Bonnet theorem

Answer

Cartan-Hadamard theorem

Answer

Nash-Moser theorem

Answer

Hopf-Rinow theorem

Question 6

Which of the following is a differential form on a 2-manifold?

Accepted Answer

ω = dx ∧ dy

Answer

ω = dx + dy

Answer

ω = dx ∧ dx + dy ∧ dy

Answer

ω = dx ∧ dy ∧ dz

Question 7

The Levi-Civita connection on a Riemannian manifold is characterized by being:

Accepted Answer

Both (a) and (b).

Answer

Torsion-free.

Answer

Metric compatible.

Answer

Neither (a) nor (b).

Question 8

In differential geometry, which fundamental concept refers to vectors that capture the infinitesimal directional change of a smooth curve?

Accepted Answer

Tangent vectors

Answer

Normal vectors

Answer

Curvature vectors

Answer

Geodesic vectors

Question 9

Consider a smooth curve embedded in R^3. The Frenet frame is a:

Accepted Answer

Moving orthonormal basis that characterizes the curve's orientation

Answer

Coordinate system that describes the curve's trajectory

Answer

Differential equation that governs the curve's motion

Answer

Set of vectors that define the curve's curvature

Question 10

Which of the following surfaces is generated by rotating a plane curve around an axis in its plane?

Accepted Answer

Sphere

Answer

Torus

Answer

Paraboloid

Answer

Hyperbolic paraboloid

Question 11

In differential geometry, the Gaussian curvature at a point on a surface provides information about:

Accepted Answer

The intrinsic curvature of the surface at that point

Answer

The direction of the steepest ascent on the surface

Answer

The area enclosed by a given boundary curve on the surface

Answer

The volume underneath the surface

Question 12

Which theorem in differential geometry provides a formula to calculate the curvature of a smooth curve in R^3?

Accepted Answer

Frenet-Serret Formula

Answer

Meusnier's Theorem

Answer

Gauss's Theorema Egregium

Answer

Gauss-Bonnet Theorem

Question 13

In differential geometry, the tangent space and normal space at a point on a surface are related as:

Accepted Answer

Orthogonal subspaces

Answer

Parallel subspaces

Answer

Intersecting subspaces

Answer

Identical subspaces

Question 14

The exterior derivative of a differential form is a:

Accepted Answer

Differential form of increased degree

Answer

Gradient vector field

Answer

Laplacian operator

Answer

Divergence-free vector field

Question 15

In differential geometry, a compact manifold is characterized by:

Accepted Answer

Finite volume

Answer

Infinite extent

Answer

Non-orientable surface

Answer

Simply connected topology

Question 16

What is the Gauss-Bonnet theorem primarily concerned with?

Accepted Answer

Relating the curvature of a surface to its Euler characteristic

Answer

Relating the curvature of a manifold to its Betti numbers

Answer

Relating the volume of a manifold to its sectional curvature

Answer

Relating the geodesics of a manifold to its topology

Question 17

What is the de Rham cohomology of a smooth manifold?

Accepted Answer

The cohomology of the complex of differential forms on the manifold.

Answer

The homology of the complex of differential forms on the manifold.

Answer

The Betti numbers of the manifold.

Answer

The Euler characteristic of the manifold.

Question 18

Which of the following is a vector field on a smooth manifold?

Accepted Answer

A smooth map from the manifold to its tangent bundle.

Answer

A smooth map from the tangent bundle to the manifold.

Answer

A smooth map from the manifold to a vector space.

Question 19

Which of the following is a Lie group?

Accepted Answer

A group that is also a smooth manifold.

Answer

A group that is also a topological manifold.

Answer

A group that is also a vector space.

Question 20

Consider a regular curve in R^3. What condition must hold in order for the curve to be a geodesic?

Accepted Answer

Its curvature vector is orthogonal to its tangent vector.

Answer

Its torsion vector is zero.

Answer

Its acceleration vector is perpendicular to its tangent vector.

Answer

Its velocity vector is constant.

Question 21

In differential geometry, what is the Frobenius theorem primarily used for?

Accepted Answer

Determining the conditions for the integrability of a system of partial differential equations

Answer

Computing the curvature of a Riemannian manifold

Answer

Solving the Plateau problem

Answer

Classifying Lie groups

Question 22

How is the curvature tensor related to the Riemann curvature tensor?

Accepted Answer

The Riemann curvature tensor can be obtained by contracting the curvature tensor with the metric tensor.

Answer

They are identical tensors.

Answer

The Riemann curvature tensor is symmetric, while the curvature tensor is not.

Answer

The curvature tensor is a tensor-valued 2-form, while the Riemann curvature tensor is a tensor-valued 4-form.

Question 23

What is the differential of a smooth map between two smooth manifolds?

Accepted Answer

A linear map between the tangent spaces at corresponding points

Answer

A bilinear map between the tangent spaces at corresponding points

Answer

A quadratic map between the tangent spaces at corresponding points

Answer

A cubic map between the tangent spaces at corresponding points

Question 24

Which of the following is an example of a symplectic manifold?

Accepted Answer

The cotangent bundle of a smooth manifold

Answer

The tangent bundle of a smooth manifold

Answer

The sphere S^2

Answer

The Euclidean space R^n

Question 25

What is the Hodge star operator primarily used for on a Riemannian manifold?

Accepted Answer

Establishing an isomorphism between the space of differential forms and its dual space

Answer

Establishing an isomorphism between the space of differential forms and the space of vector fields

Answer

Establishing an isomorphism between the space of vector fields and its dual space

Answer

Establishing an isomorphism between the space of differential forms of different degrees

Question 26

Which of the following is a core concept in differential geometry?

Accepted Answer

Tangent spaces and differential forms

Answer

Curves as submanifolds of Euclidean space

Answer

Surfaces as submanifolds of 3-space

Answer

Vector fields on manifolds

Question 27

Let M be a smooth manifold with tangent bundle TM. Which of the following is the differential of a smooth function f: M → R?

Accepted Answer

A linear map df: TM → R

Answer

A vector field on M

Answer

A 1-form on M

Answer

A 2-form on M

Question 28

What is the covariant derivative of a tangent vector along a curve in a Riemannian manifold?

Accepted Answer

The covariant derivative of the tangent vector along the curve

Answer

The derivative of the unit tangent vector

Answer

The second derivative of the position vector

Answer

The curvature of the osculating circle

Question 29

Which of the following is a fundamental theorem in differential geometry that relates the integral of curvature over a closed surface to the Euler characteristic of the surface?

Accepted Answer

The Gauss-Bonnet theorem

Answer

The Stokes theorem

Answer

The Poincaré-Hopf theorem

Answer

The de Rham theorem

Question 30

What is the Gauss map for a surface embedded in 3-dimensional Euclidean space?

Accepted Answer

A map from the surface to the unit sphere

Answer

A map from the unit sphere to the surface

Answer

A map from the tangent space to the normal space

Answer

A map from the normal space to the tangent space

Question 31

Which of the following is NOT a fundamental form on a smooth manifold?

Accepted Answer

Affine connection

Answer

Riemannian metric

Answer

Symplectic form

Answer

Volume form

Question 32

The differential of a smooth map between smooth manifolds is a tensor of which type?

Accepted Answer

(1, 1)

Answer

(0, 2)

Answer

Vector field

Answer

1-form

Question 33

Which condition is necessary for a smooth manifold to be orientable?

Accepted Answer

It has a non-vanishing vector field

Answer

Its tangent bundle is trivial

Answer

Its Euler characteristic is non-zero

Answer

Its dimension is even

Question 34

The Levi-Civita connection on a Riemannian manifold is:

Accepted Answer

Unique and torsion-free

Answer

Unique and flat

Answer

Unique and symmetric

Answer

Not unique and torsion-free

Question 35

Which of the following is an instance of a symplectic manifold?

Accepted Answer

Cotangent bundle of a smooth manifold

Answer

Riemannian manifold with constant curvature

Answer

Euclidean space

Answer

Sphere

Question 36

The curvature tensor of a Riemannian manifold is a type of tensor:

Accepted Answer

(2, 2)

Answer

(1, 3)

Answer

(0, 4)

Answer

(3, 1)

Question 37

The de Rham cohomology of a smooth manifold forms a:

Accepted Answer

Vector space over the real numbers

Answer

Algebra over the real numbers

Answer

Group over the real numbers

Answer

Ring over the real numbers

Question 38

In the context of a smooth manifold M with tangent bundle TM, which option correctly defines a vector field on M?

Accepted Answer

A smooth assignment of tangent vectors to each point in M, known as a smooth section of TM

Answer

A continuous map that associates each point in M with a vector in TM

Answer

A smooth map from M to TM that assigns a vector to each point

Answer

A differentiable map from M to TM that assigns a vector to each point

Question 39

The curvature tensor of a Riemannian manifold M serves primarily to measure:

Accepted Answer

The intrinsic curvature of the manifold, capturing its local geometric properties

Answer

The extrinsic curvature of the manifold, describing its curvature relative to an ambient space

Answer

The curvature of the Levi-Civita connection, a fundamental tool in Riemannian geometry

Answer

The curvature of the tangent bundle, providing insights into the geometry of the manifold's tangent spaces

Question 40

Which option accurately describes a differential form?

Accepted Answer

A smooth section of the cotangent bundle, assigning a linear map to each point on the manifold

Answer

A smooth real-valued function defined on the manifold

Answer

A vector field on the manifold, assigning a tangent vector to each point

Answer

A tensor field on the manifold, generalising the concept of vectors and differential forms

Question 41

The Hodge star operator, when applied to a differential form on a manifold, has which primary effect?

Accepted Answer

It produces the dual form, interchanging the roles of covectors and vectors

Answer

It modifies the degree of the differential form

Answer

It changes the sign of the differential form

Answer

It leaves the degree of the differential form unchanged

Question 42

For a smooth manifold to be considered a Lie group, which of the following requirements must be satisfied?

Accepted Answer

It must possess smooth multiplication and inversion operations, forming a group structure

Answer

It must have smooth multiplication and addition operations, forming a ring structure

Answer

It must have smooth addition and subtraction operations, forming an abelian group

Answer

It must have smooth multiplication and division operations, forming a field

Question 43

The de Rham cohomology of a smooth manifold M can be defined as:

Accepted Answer

The vector space of closed differential forms, representing cohomology classes

Answer

The vector space of exact differential forms, representing boundaries of other forms

Answer

The vector space of harmonic differential forms, satisfying the Laplace equation

Answer

The vector space of all differential forms on M, encompassing all possibilities

Question 44

Differential geometry finds notable applications in which area of physics?

Accepted Answer

General relativity, providing the mathematical framework for describing the geometry of spacetime

Answer

Electromagnetism, aiding in the understanding of electromagnetic fields and their interactions

Answer

Fluid dynamics, contributing to the modelling and analysis of fluid flow

Answer

Thermodynamics, facilitating the exploration of thermodynamic systems and their properties

Question 45

The Gauss-Bonnet theorem primarily relates:

Accepted Answer

The curvature and topology of a surface, establishing a connection between geometric and topological properties

Answer

The curvature and topology of a general manifold, extending the relationship beyond surfaces

Answer

The curvature and topology of a curve, describing the behaviour of curves on a surface

Answer

The topology and geometry of a surface, providing insights into their interplay