Differential Geometry: Manifolds, Tensors, Curvature

Build an intrinsic mental model of curved spaces: how manifolds are patched from charts, how tensors stay meaningful while components change, and how curvature shows up through connections, geodesics, and parallel transport loops.

Curvature is not a property of how a surface sits in $\mathbb{R}^3$. It is a property you can detect while living on the surface, using only distances, angles, and how directions change when you move them around. That intrinsic viewpoint is what makes differential geometry useful in physics, optimization, and data geometry: you learn which statements survive a change of coordinates and which are artifacts of your parameterization.

Curvature you can feel without $\mathbb{R}^3$

If you want to tell a sphere from a saddle intrinsically, you do not need a 3D picture. You need a rule for what counts as straight, a way to measure angles, and a way to compare directions at different points.

Three classic probes are:

• Geodesic triangles: add the three interior angles and see whether the sum is bigger, smaller, or equal to $\pi$.
• Parallel transport: carry a vector around a loop and check whether it returns rotated.
• Geodesic spreading: watch nearby geodesics converge or diverge.

Each probe is local in definition but global in effect. A tiny loop can already reveal curvature through a small rotation, and the sign of that rotation matches what you expect from spherical versus saddle behavior.

See how these intrinsic tests differ across surfaces.

A useful check is to ask what data you used. If you only used the metric notion of length and angle plus a way to compare directions along paths, you did not rely on embedding.

Intrinsic test
When a claim changes because you re-parameterize the same surface, it is about coordinates, not geometry.

Manifolds and charts that actually glue

A manifold is the promise that every point has a neighborhood that looks like $\mathbb{R}^n$ through some coordinate map. The subtlety is that one chart rarely covers the whole space without tearing or singularities, so you use an atlas.

The global object is not any single coordinate description. It is the set of charts and their overlap rules. On an overlap, two coordinate systems must be related by a smooth transition map. That map is where calculus on manifolds lives, because it tells you how objects transform.

Singularities that are not real

On the sphere, longitude becomes ill-behaved at the poles. That does not mean the manifold is singular there. It means the chart is. You fix this by adding another patch whose coordinates behave well near the poles, then ensuring the overlap transition is smooth.

Explore how patches overlap and how transition maps carry information from one chart to another.

Once you trust overlaps, you can stop worrying about where a particular coordinate formula blows up. The manifold is the thing that stays smooth while charts come and go.

Tangent and cotangent spaces as action, not arrows

A tangent vector at a point is best treated as a derivation. It is something that eats a smooth function $f$ and returns a directional derivative $v(f)$. In coordinates $(x^i)$, you can write $v=v^i\partial_i$, but the meaning is the action on functions, not the component list.

A cotangent covector is the dual kind of object. It eats a tangent vector and returns a number. Differentials like $\mathrm{d}f$ are covectors, defined by $\mathrm{d}f(v)=v(f)$. This is why gradients require a metric. Without a metric, $\mathrm{d}f$ is well-defined, but turning it into a tangent vector $\nabla f$ is not.

Watch vectors act on functions and covectors act on vectors at a point.

A common confusion is to think of covectors as vectors with subscripts. The subscript is a coordinate habit. The real difference is what the object consumes.

Duality cue
If it naturally pairs with a direction to produce a number, it belongs in the cotangent space.

Tensors are multilinear maps, not arrays

A tensor at a point is a multilinear map built from tangent and cotangent slots. A type $(k,l)$ tensor takes $k$ covectors and $l$ vectors and returns a scalar, multilinearly in each input. Components appear only after choosing a basis.

The key operational rule is this. The tensor is invariant, while its components transform to compensate for a change of basis. That is the whole point of tensor calculus: you can compute in coordinates and still represent coordinate-free facts.

Why components move when you change coordinates

Vectors and covectors transform oppositely. If $v=v^i\partial_i$, the $v^i$ change with the Jacobian of the coordinate change. For a covector $\alpha=\alpha_i\mathrm{d}x^i$, the $\alpha_i$ change with the inverse Jacobian. A general tensor mixes these behaviors across its slots.

Compare how the same tensor looks in two bases and how index positions track the transformation law.

This is also where the phrase tensor versus not a tensor lives. If a quantity does not transform by the right multilinear rule, it may be a useful bookkeeping device, but it is not a tensorial object.

Metric and Levi-Civita connection as separate jobs

A metric $g$ is a symmetric, nondegenerate $(0,2)$ tensor that lets you measure lengths and angles, and identify vectors with covectors via lowering an index. For a vector $v$, the covector $v^\flat$ is defined by $v^\flat(\cdot)=g(v,\cdot)$, and raising uses the inverse metric.

A Levi-Civita connection $\nabla$ tells you how to differentiate vector fields along directions on a manifold. It is the unique connection that is compatible with the metric, meaning $\nabla g=0$, and torsion-free.

Featured snippet view: The metric measures. The connection differentiates. Curvature comes from how covariant derivatives fail to commute.

In coordinates, the Christoffel symbols $\Gamma^k_{ij}$ encode the connection relative to a chosen chart. They are not tensor components, which is why they can be made to vanish at a point by a good coordinate choice, even on a curved manifold. What cannot be transformed away are the curvature invariants derived from them.

Explore how the metric and connection play different roles in computations.

Keep the separation clean. If you find yourself defining straight lines using only the metric, you are skipping a step. Straightness is a statement about the connection.

Geodesics and parallel transport as straightness

A geodesic is a curve $\gamma(t)$ whose velocity transports itself. Formally, it satisfies

$$\nabla_{\dot{\gamma}}\dot{\gamma}=0.$$

This equation depends on the connection. With the Levi-Civita connection, geodesics also locally extremize distance, so straightest and shortest align.

Parallel transport solves $\nabla_{\dot{\gamma}}V=0$ along a curve. It is the cleanest way to compare tangent vectors at different points without embedding. The payoff shows up on loops. In flat space, transporting around a loop brings a vector back unchanged. On curved space, you typically get a rotation whose size relates to enclosed curvature.

Try changing initial velocity and loop size to see how geodesics and transported vectors respond.

One practical way to remember what curvature does is to track failure of closure. You set out to keep a direction constant and geometry refuses, by an amount that accumulates with the loop.

Curvature tensors you can compute and interpret

Riemann curvature is the tensor that measures the noncommutativity of covariant derivatives. Acting on a vector field $Z$ with directions $X,Y$,

$$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.$$

From this, you can extract more compressed summaries.

Sectional, Ricci, scalar

• Sectional curvature assigns a number to a 2-plane in the tangent space. It matches the sphere-versus-saddle intuition in a specific direction pair.
• Ricci curvature is a trace of Riemann that measures average geodesic focusing in many directions. It is what appears in Einstein’s equations.
• Scalar curvature is the trace of Ricci. It is a single number per point, a coarse summary that still captures volume distortion trends.

Visualize how contractions reduce information from Riemann to Ricci to scalar and what each still controls.

If you are unsure which curvature to use, ask what data your question depends on. Deviations of nearby geodesics point you toward Riemann or sectional curvature. Volume growth and concentration phenomena often depend on Ricci.

Choosing the viewpoint that makes problems easier

Embedded pictures are excellent for intuition, then they quietly tempt you into using the ambient space as a crutch. When a statement truly belongs to the manifold, you should be able to rewrite it using $g$, $\nabla$, and tensors that transform correctly under chart changes.

A good next step is to take any familiar computation, maybe gradient descent on a constraint surface or motion on a sphere, and rewrite each ingredient explicitly. Which parts only need $g$? Which require $\nabla$? Where does curvature enter as a measurable obstruction?