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Question 1. Which of the following conditions is required for a topological space to be a smooth manifold? A) Compactness B) Hausdorff and second countable C) Connectedness D) Simply-connected Answer: B Explanation: A smooth manifold must be a Hausdorff space with a countable basis (second countable) to admit a countable atlas of charts. Question 2. In a chart ((U,\phi)) of an (n)-dimensional manifold, the map ( phi) takes points of (U) to which space? A) (\mathbb{R}^{n+1}) B) (\mathbb{R}^{n}) C) (\mathbb{C}^{n}) D) The unit sphere (S^{n}) Answer: B Explanation: By definition, a chart maps an open set of the manifold diffeomorphically onto an open subset of (\mathbb{R}^{n}). Question 3. Two overlapping charts ((U,\phi)) and ((V,\psi)) are said to be (C^{\infty})-compatible if the transition map (\psi\circ\phi^{-1}) is: A) Continuous B) Differentiable only once C) Smooth (infinitely differentiable) D) Analytic Answer: C
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Explanation: Compatibility for a smooth structure requires the transition map to be a smooth diffeomorphism between open subsets of (\mathbb{R}^{n}). Question 4. Which of the following manifolds is not orientable? A) The 2-sphere (S^{2}) B) The torus (T^{2}) C) The Möbius strip D) Real projective plane (\mathbb{R}P^{2}) Answer: D Explanation: (\mathbb{R}P^{2}) is non-orientable; the Möbius strip also fails orientability, but the question asks for a closed manifold, making ( mathbb{R}P^{2}) the correct answer. Question 5. The tangent space (T_{p}M) at a point (p) can be defined as the set of derivations at (p). A derivation is a linear map (D:C^{\infty}(M)\to mathbb{R}) satisfying: A) (D(f+g)=Df + Dg) only B) (D(fg)=f(p)Dg + g(p)Df) C) Both A and B D) Neither A nor B Answer: C Explanation: Derivations are linear and obey the Leibniz rule (D(fg)=f(p)Dg+g(p)Df). Question 6. Which description correctly characterizes an immersion (f:M\to N)? A) The differential (df_{p}) is surjective for all (p\in M). B) The differential (df_{p}) is injective for all (p\in M).
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A) Acting on vectors by matrix multiplication. B) (df_{p}(X)(g)=X(g\circ f)) for all (g\in C^{\infty}(N)). C) Pulling back covectors. D) None of the above. Answer: B Explanation: The push-forward sends a derivation (X) at (p) to a derivation at (f(p)) by composing test functions with (f). Question 10. The tangent bundle (TM) of a smooth manifold (M) is itself a smooth manifold of dimension: A) (\dim M) B) (2\dim M) C) (\dim M+1) D) (\dim M-1) Answer: B Explanation: Each fiber (T_{p}M) is an (\dim M)-dimensional vector space, and the base adds another (\dim M) dimensions, giving total (2\dim M). Question 11. Which of the following is a correct description of a vector field on a manifold (M)? A) A smooth function (M\to\mathbb{R}). B) A smooth section of the cotangent bundle (T^{*}M). C) A smooth section of the tangent bundle (TM). D) A differential 2-form on (M). Answer: C Explanation: By definition, a vector field assigns to each point a tangent vector smoothly, i.e., a section of (TM).
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Question 12. If (X) and (Y) are smooth vector fields on (M), their Lie bracket ([X,Y]) satisfies: A) ([X,Y]=[Y,X]) B) ([X,Y]= -[Y,X]) C) ([X,Y]=0) for all (X,Y). D) None of the above. Answer: B Explanation: The Lie bracket is antisymmetric: ([X,Y] = -[Y,X]). Question 13. An integral curve (\gamma(t)) of a vector field (X) satisfies which differential equation? A) (\gamma'(t)=X(\gamma(t))) B) (\gamma'(t)=\nabla X(\gamma(t))) C) (\gamma''(t)=X(\gamma(t))) D) (\gamma'(t)=\mathrm{div},X(\gamma(t))) Answer: A Explanation: By definition, an integral curve’s velocity equals the vector field evaluated at the point. Question 14. The flow (\Phi^{X}{t}) of a complete vector field (X) is a family of diffeomorphisms satisfying: A) (\Phi^{X}{0}= \text{id}{M}) and (\Phi^{X}{t+s}= \Phi^{X}{t}\circ Phi^{X}{s}). B) (\Phi^{X}{t}) is only defined for (t\ge 0). C) (\Phi^{X}{t}) is always linear. D) None of the above.
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C) (\alpha\wedge\beta = 0) if (p\neq q) D) None of the above. Answer: B Explanation: The wedge product is graded-commutative; swapping introduces a sign ((-1)^{pq}). Question 18. Which of the following forms on (\mathbb{R}^{3}) is closed but not exact? A) (dx) B) (x,dy) C) (\frac{x,dy - y,dx}{x^{2}+y^{2}}) (restricted to (\mathbb{R}^{3} setminus{z\text{-axis}})) D) (dz) Answer: C Explanation: This form is the angular 1-form on the punctured plane; it is closed (its exterior derivative vanishes) but not exact because its integral around a loop encircling the axis is non-zero. Question 19. The Poincaré Lemma asserts that on a contractible open set (U subset\mathbb{R}^{n}): A) Every closed form is exact. B) Every exact form is closed. C) Both A and B hold, but only for 1-forms. D) None of the above. Answer: A Explanation: In a star-shaped (hence contractible) domain, any closed differential form is the exterior derivative of another form.
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Question 20. A smooth manifold (M) with boundary (\partial M) has which of the following properties? A) (\partial M) is itself a smooth manifold without boundary of dimension (\dim M- 1). B) (\partial M) is empty for all manifolds. C) The interior of (M) is not a manifold. D) None of the above. Answer: A Explanation: The boundary inherits a smooth structure of one lower dimension and has no boundary of its own. Question 21. In the definition of an oriented atlas, the Jacobian determinant of each transition map must be: A) Positive everywhere on the overlap. B) Negative everywhere on the overlap. C) Non-zero but may change sign. D) Zero. Answer: A Explanation: Orientation requires that all transition maps preserve orientation, i.e., have positive Jacobian determinant. Question 22. Which of the following is a correct statement of the Generalized Stokes’ Theorem? A) (\displaystyle\int_{M} d\omega = \int_{\partial M}\omega) for any compact oriented (M) and (\omega\in\Omega^{\dim M-1}(M)). B) (\displaystyle\int_{\partial M} d\omega = \int_{M}\omega). C) (\displaystyle\int_{M} \omega = 0) for all exact (\omega).
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B) (df_{p}) is surjective. C) (df_{p}) is an isomorphism. D) (f) is a homeomorphism onto its image. Answer: B Explanation: Submersions have surjective differentials, giving locally the structure of a projection. Question 26. The preimage theorem states that if (f:M\to N) is a smooth submersion and (c\in N) is a regular value, then (f^{-1}(c)) is: A) Empty. B) A smooth submanifold of (M) of codimension (\dim N). C) A discrete set of points. D) None of the above. Answer: B Explanation: The preimage of a regular value under a submersion is a submanifold whose codimension equals the dimension of the target. Question 27. Which of the following best describes a partition of unity subordinate to an open cover ({U_{\alpha}}) of a manifold (M)? A) A collection of smooth functions ({\phi_{\alpha}}) with (\sum_{\alpha}\phi_{ alpha}=1) everywhere and (\operatorname{supp}\phi_{\alpha}\subset U_{ alpha}). B) A set of disjoint open subsets covering (M). C) A finite set of charts covering (M). D) None of the above. Answer: A Explanation: Partitions of unity allow gluing local data into global objects while respecting the cover.
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Question 28. On the torus (T^{2}=S^{1}\times S^{1}), the 1-form ( theta_{1}=d\theta) (first circle coordinate) is: A) Exact. B) Closed but not exact. C) Neither closed nor exact. D) Zero. Answer: B Explanation: (\theta_{1}) has integral (2\pi) over the first circle, so it cannot be exact, yet (d\theta_{1}=0), making it closed. Question 29. Which of the following is true about the Lie derivative ( mathcal{L}{X}) acting on a differential form (\omega)? A) (\mathcal{L}{X}\omega = d(\iota_{X}\omega) + \iota_{X}d\omega). B) (\mathcal{L}{X}\omega = \iota{X}d\omega - d(\iota_{X}\omega)). C) (\mathcal{L}_{X}\omega = d\omega). D) None of the above. Answer: A Explanation: Cartan’s formula expresses the Lie derivative in terms of interior product and exterior derivative. Question 30. A smooth map (f:M\to N) is called a diffeomorphism if: A) It is bijective and both (f) and (f^{-1}) are continuous. B) It is bijective and both (f) and (f^{-1}) are smooth. C) Its differential is invertible at every point. D) None of the above.
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C) (U\cap V) itself. D) The whole manifold (M). Answer: A Explanation: The transition map takes points expressed in (\phi)-coordinates to ( psi)-coordinates, thus its domain is (\phi(U\cap V)). Question 34. The differential of the inclusion map (i:S^{1}\hookrightarrow mathbb{R}^{2}) at a point (p\in S^{1}) is: A) An isomorphism from (T_{p}S^{1}) to (\mathbb{R}^{2}). B) The zero map. C) An injection identifying (T_{p}S^{1}) with a line in (\mathbb{R}^{2}). D) None of the above. Answer: C Explanation: The inclusion’s differential embeds the 1-dimensional tangent space as a line (the tangent line) in (\mathbb{R}^{2}). Question 35. Which of the following is a necessary condition for a smooth map (f:M\to N) to be a local diffeomorphism at (p\in M)? A) (df_{p}) is injective. B) (df_{p}) is surjective. C) (df_{p}) is an isomorphism. D) (f) is globally bijective. Answer: C Explanation: The Inverse Function Theorem states that if (df_{p}) is invertible, then (f) is a diffeomorphism between neighborhoods of (p) and (f(p)).
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Question 36. The volume form on an oriented Riemannian manifold ((M,g)) can be expressed locally as: A) (\sqrt{\det(g_{ij})},dx^{1}\wedge\cdots\wedge dx^{n}). B) (\det(g_{ij}),dx^{1}\wedge\cdots\wedge dx^{n}). C) (dx^{1}\wedge\cdots\wedge dx^{n}) irrespective of (g). D) None of the above. Answer: A Explanation: The Riemannian volume form involves the square root of the determinant of the metric matrix. Question 37. If (\omega) is a compactly supported (n)-form on an oriented (n)-manifold (M) without boundary, then (\displaystyle\int_{M}d\omega) equals: A) (\displaystyle\int_{\partial M}\omega). B) Zero, because (\partial M=\emptyset). C) The Euler characteristic of (M). D) None of the above. Answer: B Explanation: Since (\partial M=\emptyset), Stokes’ theorem gives (\int_{M}d omega=0). Question 38. The Euler characteristic (\chi(M)) of a compact oriented surface can be computed using de Rham cohomology as: A) (\displaystyle\sum_{k=0}^{2}(-1)^{k}\dim H^{k}{\text{dR}}(M)). B) (\displaystyle\sum{k=0}^{2}\dim H^{k}{\text{dR}}(M)). C) (\displaystyle\dim H^{1}{\text{dR}}(M)). D) None of the above. Answer: A
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C) ((f^{}\alpha){p}= \alpha{p}\circ f). D) None of the above. Answer: A Explanation: Pullback composes a covector with the differential of the map. Question 42. On a smooth manifold (M), a function (f) is called a Morse function if: A) All its critical points are non-degenerate. B) Its Hessian vanishes everywhere. C) It has no critical points. D) None of the above. Answer: A Explanation: Morse functions have isolated, non-degenerate critical points, crucial for Morse theory. Question 43. The Hodge star operator () on an oriented Riemannian (n)- manifold maps a (k)-form to an: A) ((n-k))-form. B) (k)-form. C) ((k+1))-form. D) None of the above. Answer: A Explanation: The Hodge star provides an isomorphism between ( Lambda^{k}T^{}M) and (\Lambda^{n-k}T^{}M). Question 44. In the context of fiber bundles, a section of a bundle (E\to M) is:
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A) A smooth map (s:M\to E) with (\pi\circ s=\text{id}{M}). B) A smooth map (s:E\to M) that is a left inverse of (\pi). C) A diffeomorphism between (E) and (M). D) None of the above. Answer: A Explanation: A section picks a point in each fiber smoothly, satisfying the projection condition. Question 45. The exponential map (\exp{p}:T_{p}M\to M) of a Riemannian manifold is defined using: A) Geodesics emanating from (p). B) The Lie group structure of (M). C) The embedding of (M) into Euclidean space. D) None of the above. Answer: A Explanation: (\exp_{p}(v)) is the endpoint at time 1 of the geodesic starting at (p) with initial velocity (v). Question 46. Which of the following manifolds is parallelizable? A) The 2-sphere (S^{2}). B) The 3-sphere (S^{3}). C) The real projective plane (\mathbb{R}P^{2}). D) None of the above. Answer: B Explanation: (S^{3}) admits a global frame (e.g., via identification with the Lie group (SU(2))), while (S^{2}) does not.
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D) The space of locally constant functions, which equals (\mathbb{R}^{# text{components}}). Answer: D Explanation: Closed 0-forms are locally constant; their equivalence classes correspond to the number of connected components. Question 50. For a smooth map (f:\mathbb{R}^{2}\to\mathbb{R}) given by (f(x,y)=x^{2}+y^{2}), the level set (f^{-1}(1)) is: A) A smooth 1-dimensional submanifold (circle). B) A single point. C) The whole plane. D) None of the above. Answer: A Explanation: The set ({(x,y)\mid x^{2}+y^{2}=1}) is the unit circle, a 1 - dimensional embedded submanifold. Question 51. Which of the following is a correct statement of the Inverse Function Theorem for manifolds? A) If (df_{p}) is invertible, then there exist neighborhoods (U\ni p) and (V\ni f(p)) such that (f|{U}:U\to V) is a diffeomorphism. B) If (df{p}) is injective, then (f) is globally invertible. C) If (df_{p}) is surjective, then (f) has a smooth inverse near (p). D) None of the above. Answer: A Explanation: The theorem guarantees a local diffeomorphism when the differential is a linear isomorphism.
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Question 52. In the definition of a smooth manifold with boundary, a chart ((U, phi)) mapping to (\mathbb{H}^{n}={x\in\mathbb{R}^{n}\mid x^{n}\ge0}) must satisfy: A) (\phi(U\cap\partial M)\subset{x^{n}=0}). B) (\phi(U)\subset\mathbb{R}^{n}) (no half-space). C) (\phi) is a homeomorphism onto an open set of (\mathbb{R}^{n}). D) None of the above. Answer: A Explanation: Points of the boundary correspond to points mapped to the hyperplane (x^{n}=0) in the half-space chart. Question 53. The term “regular value” of a smooth map (f:M\to N) means: A) The differential (df_{p}) is zero for all (p) in the preimage. B) For every (p\in f^{-1}(c)), (df_{p}) is surjective. C) The level set (f^{-1}(c)) is empty. D) None of the above. Answer: B Explanation: Regular values have surjective differentials at all points of their preimage, ensuring nice submanifold structure. Question 54. Which of the following manifolds has non-trivial first de Rham cohomology group? A) The 2-sphere (S^{2}). B) The torus (T^{2}). C) The 3-sphere (S^{3}). D) All of the above. Answer: B
