Differential geometry becomes much easier once you can manufacture your own examples. The field looks forbidding when you only encounter examples as finished objects: the sphere, hyperbolic space, Lie groups, complex projective space. The craft is in the constructions and in the habit of checking a few invariants that tell you what you actually built.
This article is a practical recipe for building examples you can compute with, ranging from gentle to serious. The goal is not to list every construction, but to give a compact toolkit that repeatedly produces manifolds, metrics, connections, and curvature calculations without guesswork.
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The minimal example-building toolbox
Almost every example you will meet is assembled from a small set of moves:
- Start from a model space with known geometry.
- Modify the metric in a controlled way.
- Take a quotient by a group action when you want topology.
- Form a submanifold or a bundle when you want constraints.
- Glue pieces together when you want global features not visible locally.
Each move comes with a standard set of checks. If you do those checks consistently, you can build examples with confidence instead of relying on memorized templates.
Step zero: choose the object class and the invariant you want
Before building anything, decide what kind of object you are constructing and what feature you are trying to control.
A useful planning table:
| Object you want | Typical input | Typical invariant you control |
|—|—|—|
| Riemannian manifold $(M,g)$ | smooth $M$ and metric $g$ | curvature, geodesics, volume growth |
| Connection $\nabla$ | vector bundle and local connection forms | holonomy, parallel transport |
| Submanifold | embedding or immersion | second fundamental form, induced curvature |
| Quotient space | group action $G \curvearrowright M$ | fundamental group, orbifold vs manifold |
| Fiber bundle | structure group + transition functions | characteristic classes, curvature of connection |
If you skip this step, you tend to produce objects that are pretty but irrelevant to the theorem you are studying.
Move A: products and warped products
The fastest way to build new manifolds is to take products.
- If $(M,g_M)$ and $(N,g_N)$ are Riemannian, the product $M \times N$ has the product metric $g_M \oplus g_N$.
- Geodesics split, curvature decomposes, and computations often reduce to the factors.
If you need curvature that changes along one direction, use a warped product. Given a positive smooth function $f: B \to \mathbb{R}_{>0}$, define on $B \times F$:
Warped products generate many standard geometries, including model cosmological metrics in mathematical physics and many examples with controlled sectional curvature bounds.
Checks to run:
- Smoothness and positivity of $f$.
- Completeness, often via growth conditions on $f$ and completeness of $B,F$.
- Curvature formulas, especially how terms involving $\nabla f$ and $\nabla^2 f$ enter.
A practical habit:
- Choose $B$ one-dimensional when learning, so $f$ depends on a single variable and the Hessian terms become ordinary derivatives.
Move B: conformal changes \to a metric
Given a Riemannian metric $g$ on $M$ and a smooth function $u$, define a conformal metric
Conformal changes preserve angles but not lengths. They are extremely useful because they turn geometry into analysis: curvature transforms by explicit formulas involving $u$ and its derivatives.
In dimension two, the transformation of Gauss curvature is especially clean:
This lets you build metrics with prescribed curvature by solving an elliptic equation.
Checks to run:
- Ensure $u$ is smooth globally, not just in a chart.
- Track the Laplacian sign convention you are using.
- Confirm completeness, because conformal factors can make distances finite where they used to be infinite.
Move C: quotients by isometries to control topology
Quotients are the most efficient way to get interesting topology while keeping geometry computable.
Start with a manifold $(\widetilde{M}, \widetilde{g})$ and a discrete group $\Gamma$ acting by isometries. If the action is:
- free, and
- properly discontinuous,
then the quotient $M = \widetilde{M}/\Gamma$ is a smooth manifold and the metric descends.
What you gain:
- The universal cover is still $\widetilde{M}$, so local geometry is inherited.
- The fundamental group is essentially $\Gamma$, so topology is explicit.
Standard families you can generate immediately:
- Flat manifolds from $\mathbb{R}^n$ and Euclidean isometry groups.
- Hyperbolic surfaces from $\mathbb{H}^2$ and Fuchsian groups.
- Constant curvature three-manifolds from $\mathbb{H}^3$ and Kleinian groups.
Checks to run:
- Confirm freeness: no nontrivial group element fixes a point.
- Confirm proper discontinuity: every compact set meets only finitely many of its translates.
- Confirm cocompactness if you want a compact quotient.
A quick diagnostic for freeness in linear examples:
- If $\Gamma$ is generated by translations, the action is free.
- If reflections or rotations appear, fixed points may exist and you may get an orbifold rather than a manifold.
Move D: submanifolds and induced geometry
Given an immersion $i: S \hookrightarrow M$, the induced metric on $S$ is simply $i^\*g$. This produces geometry constrained \to a lower-dimensional set.
The main new ingredient is extrinsic:
- The second fundamental form $II$.
- The shape operator $A$.
- The Gauss and Codazzi equations relating intrinsic curvature of $S$ \to curvature of $M$ and $II$.
A productive way to build examples:
- Choose $M$ with easy geometry, like $\mathbb{R}^n$, $S^n$, or a product.
- Choose $S$ defined by simple equations, like level sets $F=c$ or graphs of functions.
- Compute $II$ using gradients and Hessians of $F$.
This yields examples where you can see curvature emerge from embedding constraints, which is often the geometric intuition behind PDE constraints on $F$.
Checks to run:
- Verify regular value conditions so the level set is a smooth submanifold.
- Identify a clean unit normal field.
- Compute $II$ in a coordinate-free way when possible to avoid index confusion.
Move E: bundles and connections as controlled “twisting”
A vector bundle or principal bundle is the right way to encode twisting that is invisible locally.
To build a bundle:
- Choose a base manifold $B$.
- Choose a structure group $G$.
- Specify transition functions $g_{\alpha\beta}$ on overlaps of a cover.
To build a connection:
- Choose local connection 1-forms $A_\alpha$ satisfying the gauge transformation rule on overlaps.
- Compute curvature $F = dA + A \wedge A$ (or $F = dA$ in abelian cases).
This is where many global invariants live:
- holonomy,
- characteristic classes,
- and obstructions to global frames.
A concrete beginner-friendly family:
- Line bundles over $S^2$, where curvature integrates to an integer multiple of $2\pi$ (a topological invariant).
- Tangent bundle of $S^2$, where the impossibility of a nowhere-vanishing continuous tangent vector field expresses a global obstruction.
Checks to run:
- Verify cocycle conditions for transition functions.
- Verify curvature is globally well-defined under gauge transformations.
- Use Stokes-type reasoning to relate integrals of curvature to topology.
Move F: gluing and surgery for global behavior
Some phenomena are global and cannot be seen in a single chart or a simple quotient. Gluing produces them.
Common gluing moves:
- Connected sum $M \# N$, which splices manifolds along deleted balls.
- Gluing along boundaries with an explicit diffeomorphism.
- Doubling a manifold with boundary to remove the boundary.
For Riemannian metrics, gluing is subtle because smoothness and curvature control require transition regions. A useful technique is:
- Choose metrics that are product-like near the gluing boundary.
- Insert a collar where you interpolate using a smooth cutoff.
Checks to run:
- Ensure the gluing diffeomorphism matches orientations when needed.
- Ensure metric interpolation keeps the metric positive definite.
- Track how curvature changes in the interpolation region, since that is where curvature is created.
Gluing is often where one learns the difference between “topological existence” and “geometric existence with bounds.”
A worked construction thread: building a family with computable curvature
Here is a practical construction you can carry out repeatedly.
Start with the product $S^1 \times \mathbb{R}$ with coordinates $(\theta, r)$. Define a metric
where $f(r) > 0$ is smooth.
This is a metric coming from a surface generated by rotation, but you do not need to embed it in $\mathbb{R}^3$ \to compute.
The Gauss curvature for such a metric is
Now you can choose $f$ \to manufacture curvature:
- If $f(r) = 1$, then $K=0$ and you get a flat cylinder.
- If $f(r) = \cosh(r)$, then $K=-1$ and you get a hyperbolic-type metric in suitable coordinates.
- If $f(r) = \sin(r)$ on an interval, you get positive curvature like the sphere in polar coordinates.
The same template lets you build complete metrics or incomplete metrics depending on the behavior of $f$ at the ends, and you can see completeness by analyzing the length of curves heading to the ends.
This single construction captures the core example-building skill:
- choose a simple ansatz,
- compute the invariant in terms of the ansatz,
- pick the function to force the invariant you want.
How to avoid “example drift”
When you build examples, it is easy to drift into unrelated features. A discipline that keeps you on topic is to write, for every example, a short “invariant report”:
- Topology: connected, compact, fundamental group if simple.
- Metric: complete or not, volume growth if relevant.
- Curvature: sign, boundedness, constant vs variable.
- Geodesics: any closed geodesics, any conjugate points.
- Symmetry: isometry group or at least obvious Killing fields.
If your example cannot be summarized this way, you probably do not yet know what you built, and computations downstream will become unreliable.
Summary: a repeatable workflow
A repeatable differential geometry example workflow is:
- Pick the feature you want to control.
- Select a move: product, conformal change, quotient, submanifold, bundle, gluing.
- Run the standard checks: smoothness, freeness of action, completeness, curvature formulas.
- Produce a short invariant report to lock in what the example actually is.
Once you can do this, reading differential geometry papers becomes easier because you start recognizing which move the author is using and what invariants they are trying to force. Example-building is not extra. It is the engine that makes the subject feel navigable.
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