Riemannian geometry is an advanced and fascinating field of mathematics with applications across science and engineering. As a job candidate with background in this area, you can expect technical interviews to contain numerous Riemannian geometry questions testing your conceptual grasp and problem-solving abilities.
In this article we provide an in-depth look at 25 typical Riemannian geometry interview questions. For each question, we explain the key concepts involved and provide tips on how to approach the problems. Our goal is to help prepare you for success in landing your dream role.
Fundamentals of Riemannian Geometry
Q1: Can you explain the fundamental concepts of Riemannian geometry?
Riemannian geometry studies smooth manifolds equipped with Riemannian metrics. A Riemannian metric defines an inner product on each tangent space that varies smoothly from point to point. This allows for computation of lengths, angles, curvature, geodesics (shortest paths), etc. Key concepts include the metric tensor, Levi-Civita connection, curvature tensor, and geodesic equations. Grasping these fundamentals is essential.
Q2: How is the Riemannian metric defined and used in this field?
The Riemannian metric is a smoothly varying inner product on the tangent bundle of a manifold. It enables measurement of lengths and angles, calculation of distances, areas, volumes and curvature. It generalizes Euclidean geometry to curved spaces, providing tools to understand intrinsic properties. Metrics are fundamental objects of study, enabling key computations.
Q3: Can you describe the key differences between Euclidean and Riemannian geometry?
Euclidean geometry deals with flat spaces and shapes in 2 or 3 dimensions, with parallel lines never intersecting. Riemannian geometry generalizes this to study curved manifolds in any dimension, introducing ideas like the metric tensor and curvature. While Euclidean geometry is limited to flat spaces, Riemannian geometry can handle curved spaces, broadening the scope.
Applications of Riemannian Geometry
Q4: How can Riemannian geometry be applied to solve problems in data science or machine learning?
Riemannian geometry provides tools like geodesic distances to analyze complex high-dimensional data by mapping it to lower-dimensional manifolds. This simplifies structure, enabling better analysis. In ML, it helps optimize algorithms on curved spaces like Stiefel manifolds. Riemannian optimization navigates these spaces efficiently, improving performance.
Q5: How would you use Riemannian geometry to map and explore complex data structures?
The Riemannian framework allows understanding high-dimensional data geometry using concepts like distances and curvature Techniques like manifold learning perform dimensionality reduction, while geodesic distances can cluster data points Riemannian geometry is ideal for data with inherent curved structure, like in computer vision or medical imaging.
Q6: How does the concept of Riemannian manifold contribute to shape analysis in computer vision?
Riemannian manifolds enable modeling shapes as points in high-dimensional space. Their intrinsic geometry captures essential shape characteristics like curvature. This provides a robust framework for shape comparison and classification. Riemannian metrics quantify shape similarity. Overall, they provide a powerful toolkit for shape analysis.
Curvature and Geodesics
Q7: How is the concept of curvature used and interpreted in Riemannian geometry?
Curvature quantifies deviation from Euclidean geometry. The curvature tensor provides a quantitative measure. Positive Ricci curvature indicates regions smaller than Euclidean, while negative values imply larger regions. Weyl tensor measures tidal forces. Scalar curvature describes overall curvature. Gauss-Bonnet links total curvature with topology. Curvature is a central concept revealing deep connections between geometry and topology.
Q8: Can you explain the concept of geodesics and their significance in Riemannian geometry?
Geodesics are shortest paths between points on a curved manifold, generalizing straight lines in Euclidean space. They represent paths of least energy for particles. In general relativity, geodesics are trajectories of free-falling objects in spacetime. As locally distance-minimizing curves, geodesics are fundamental to intrinsic Riemannian geometry. The geodesic equation governs their behavior.
Q9: How would you compute the shortest path between two points on a curved surface using Riemannian geometry?
Using the Riemannian metric, we can compute geodesics as shortest paths via the geodesic equation. This second order differential equation is derived from the Euler-Lagrange equations. With initial point and direction specified, numerical methods like Runge-Kutta approximate solutions. Globally minimizing paths may not exist due to manifold topology. Local techniques suffice.
Curvature Relationships and Comparison Theorems
Q10: Can you explain how sectional, Ricci, and scalar curvatures are interconnected?
These curvatures derive from the curvature tensor, providing different insights. Scalar curvature is the trace of Ricci, giving average manifold curvature. Ricci focuses on volume distortion. Sectional curvature quantifies curvature on 2D tangent space slices. Together, they fully characterize local geometry, relating to topology and geodesic behavior.
Q11: What role does the Riemann curvature tensor play in Riemannian geometry?
The Riemann tensor provides the core measure of intrinsic curvature, quantifying the failure of parallel transport to be Euclidean. It defines geodesics and appears in the field equations of general relativity, where it describes spacetime curvature due to mass-energy. As the foundational curvature measure, it’s indispensable for understanding manifold geometry.
Q12: How would you explain sectional curvature and its usage in Riemannian geometry?
Sectional curvature measures how much geodesics emanating from a point diverge from Euclidean space within a 2D tangent space section. Positive values indicate spherical geometry, negative imply hyperbolic. Zero is locally Euclidean. Crucially, sectional curvature reveals local geometric and topological properties. It features in key results like Gauss-Bonnet and comparison theorems.
Connections to Physics and General Relativity
Q13: How does Riemannian geometry contribute to understanding spacetime structure in physics?
Riemannian geometry provides the mathematical framework underpinning general relativity. Metrics model gravitational fields, with curvature encoding the effects of mass-energy on spacetime geometry. This allowed Einstein to revolutionize our conception of gravity as geometry rather than force. Riemannian geometry was essential for this profound insight.
Q14: Can you explain the relation between Riemannian geometry and general relativity?
In general relativity, gravity is viewed as spacetime curvature rather than force. Mass and energy curve spacetime, with free particles following geodesics. Riemannian geometry provides the language to formulate this idea mathematically. The metric tensor encodes gravity; the Riemann tensor describes curvature; geodesics are free-fall paths. This nexus enabled Einstein’s breakthrough theory.
Q15: How are scalar, sectional, and Ricci curvatures interconnected in general relativity?
These curvatures all derive from the Riemann tensor, which quantifies spacetime curvature in general relativity. Scalar curvature relates to volume; Ricci describes tidal forces and matter distribution; sectional characterizes local geometry. Together, they encode gravity’s effect on rods, clocks, matter, and light for the Einstein field equations.
Advanced Concepts
Q16: What is the Ricci flow and how can it be applied in Riemannian geometry?
The Ricci flow deforms the metric analogous to heat diffusion, with equation dg/dt = -2Ric. It was used by Perelman to prove the Poincaré conjecture. Ricci flow allows understanding manifold structure by smoothing out irregularities. It enables classification based on topology and helps prove profound results.
Q17: Can you explain the concept of parallel transport in Riemannian geometry?
Parallel transport preserves vector direction along smooth curves. It’s defined using the Levi-Civita connection, which gives a unique solution to the differential equation specifying parallel transport. The curvature tensor quantifies how parallel transport around a loop deviates from identity. Parallel transport is foundational for defining manifold geometry.
Q18: What are the connections between Riemannian geometry and other mathematical fields?
Riemannian geometry connects deeply with fields like algebra through Lie groups, analysis via geodesics and harmonic functions, and complex geometry through Kähler manifolds. The unifying nature of Riemannian geometry and its interactions with these subjects has led to major cross-pollination of ideas throughout mathematics.
Open Problems
Q19: Can you discuss some open problems or unsolved questions in Riemannian geometry?
Major open problems include the Hopf conjecture on sectional curvature and diffeomorphism type of even-dimensional manifolds, the positive mass conjecture relating total mass and manifold curvature, and the existence of Einstein metrics on compact manifolds. Solving these problems would lead to profound advances in our geometric understanding.
Q20: What are some “embarrassingly simple to state” open problems that capture the essence of Riemannian geometry?
Some such problems are: does every metric on S3 have infinitely many prime closed geodesics? Do there exist metrics on S
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Lots of surveys and books have problems that are still open. It would be helpful to have a list of twelve problems that are still open but so easy to state. A list that is “folklore” and that every graduate student in differential geometry should keep in his/her pocket.
Here are the ones I like best:
1. Does every Riemannian metric on the $3$-sphere have at least three prime closed geodesics? If not, does it have an infinite number of them?
2. If the volume of a Riemannian $3$-sphere is 1, does it have a closed geodesic whose length is less than $10^{24}$? If you like $S^1 times S^2$ more, ask the same thing.
3. Does $S^2 times S^2$ admit a Riemannian metric with positive sectional curvature?
4. The canonical metric and a Riemannian metric on real projective space have the same volume. Does the Riemannian metric carry a closed, non-contractible geodesics whose length is at most $pi$?
5. How do you solve the isoperimetric problem in the complex projective plane given its standard metric (Fubini-Study)?
6. Is the canonical metric on the complex projective plane the only Riemannian metric on this manifold that does not allow geodesics to be open?
7. Is there a closed geodesic on the $2$-sphere that has a length of no more than $2pi$ and a Riemannian metric that is close enough to the canonical metric? There should be one.
The book “A Panoramic View of Riemannian Geometry” by Marcel Berger includes a number of open problems.
There is also “Review of Geometry and Analysis” by S.-T. Yau (Asian Journal of Mathematics, vol. 4, no. 1, pp. 235-278, March 2000), where he discusses many big open problems in Riemannian geometry, symplectic geometry, algebraic geometry, and geometric analysis. This can keep you occupied for a long long time…
Here are two possible relevant references, one from 1998 and the other from 2008, but neither of which I can confidently evaluate:
(1) Thierry Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, 1998.
(2) Simon Donaldson, “Some problems in differential geometry and topology,” Nonlinearity 21 T157, 2008.
Here is one sentence from Donaldsons paper:
Gromovs “Spaces and Questions” sketches some big themes and associated questions in Geometry. Atiyahs lectures discuss themes inspired by physics.
One of my favourite open problems is this:
Are there any examples of irreducible, compact, simply-connected, riemannian manifolds with vanishing Ricci curvature with generic (i. e. , not special) holonomy?.
Just to make this answer self-contained, I pasted the content from the wiki article here: http://en.wikipedia.org/wiki/Shing-Tung_Yau#Open_problems
Yau has compiled an influential set of open problems in geometry.
- Harmonic functions with controlled growth
One of Yau’s problems is about bounded harmonic functions, and harmonic functions on noncompact manifolds of polynomial growth. He first showed that there are no bounded harmonic functions on manifolds with positive curvatures. Then he came up with the Dirichlet problem at infinity for bounded harmonic functions on negatively curved manifolds. Finally, he moved on to harmonic functions of polynomial growth. The author, Dennis Sullivan, talks about Yaus’s geometric intuition and how it caused him to turn down Sullivan’s analytical proof. Michael Anderson independently came to the same conclusion about the bounded harmonic function on negatively curved manifolds that are simply connected and use a geometric convexity construction.
- Rank rigidity of nonpositively curved manifolds
Based on Mostow’s strong rigidity theorem again, Yau asked for a concept of rank for general manifolds that goes beyond the one for locally symmetric spaces. He also asked for rigidity properties for higher rank metrics. Progress has been made in this area by Ballmann, Brin, and Eberlein’s work on non-positive curved manifolds, Gromovs and Eberlein’s metric rigidity theorems for higher rank locally symmetric spaces, and Ballmann and Burns-Spatzier’s classification of closed higher rank manifolds with non-positive curvature. This leaves rank 1 manifolds of non-positive curvature as the focus of research. They behave more like manifolds of negative curvature, but remain poorly understood in many regards.
- Kähler–Einstein metrics and stability of manifolds
It is known that if a complex manifold has a Kähler–Einstein metric, then its tangent bundle is stable. In the early 1980s, Yau realized that the fact that Kähler manifolds have special metrics is the same thing as the manifolds being stable. Various people including Simon Donaldson have made progress to understand such a relation.
In the past, he has worked with string theorists like Strominger, Vafa, and Witten, as well as theoretical physicists like B. Greene, E. Zaslow and A. Klemm . The Strominger–Yau–Zaslow program is to construct explicitly mirror manifolds. David Gieseker wrote that the Calabi conjecture was a key idea in connecting string theory and algebraic geometry. It was especially important for the progress made in the SYZ program, the mirror conjecture, and the Yau–Zaslow conjecture.
Here are two more to add to the listed above:
Conjecture of LeBrun and Salamon : Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?
Some open problems can be found in the last part of Shing-Tung Yaus’s book Seminar on differential geometry. It is called the Problem section.
You can try one of these: http://www.aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. All of them concern with nonnegatively curved Riemannian manifolds and Alexandrov geometry. In the same context I know a couple of surveys: http://arxiv.org/abs/0707.3091 and http://arxiv.org/abs/math/0701389. It has been conjectured (you can check in those papers) that any nonnegatively curved manifold is rationally elliptic. This is an important open problem in Riemannian geometry.
AIM maintains a list of open problems from workshops that it hosts. You could try looking there (but they may be too specific for your needs).
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Riemannian Geometry – Definition: Oxford Mathematics 4th Year Student Lecture
FAQ
What is Riemannian geometry used for?
What is the connection in Riemannian geometry?
What is the fundamental theorem of Riemannian geometry?
How is Riemannian geometry different from Euclidean geometry?
What are the applications of Riemannian geometry?
Applications Riemannian geometry appears in many areas of pure and applied mathematics (e.g., minimal surfaces, curvature flows optimizing shapes, learning) as well as in mathematical physics (e.g., Einstein’s theory of relativity, noncommutative geometry).
What is Riemannian geometry?
Riemannian geometry is the branch of diferential geometry, where the ancient geometric objects such as length, angles, areas, volumes and curvature come back to life in a modern reincarnation on smooth manifolds. Read [36, Ch. 1] for an intuitive understanding of curvature and to understanding where we are aiming at with this course.
How do you test a Riemannian conjecture?
Hence [m, m] h. If G is compact, then there are (G G)-invariant metrics, and you can show they have nonnegative sectional curvature. These kinds of spaces, like homogeneous Riemannian manifolds in general, have nice formulas for their curvature, and hence are an excellent playground for testing conjectures in Riemannian geometry.
Are Riemannian geometry lectures original?
These lectures are entirely expository and no originality is claimed. Where necessary, references are indicated in the text. Collapse in Riemannian geometry is the phenomenon of injectivity radii limiting to zero, while sectional curvatures remain bounded.
