A curve is more than its equation. A surface is more than its picture. Differential geometry asks what can be measured without confusing the shape with the coordinates used to describe it.
Imagine marking a route on a cylinder, then cutting the cylinder along a vertical line and unrolling it. The route looks different on the flat sheet, yet its length need not change. Now try the same operation with a spherical shell. Flattening it without stretching is a different problem. The difference is not explained by saying that both objects look curved. We need to distinguish the bending of a surface in space from the geometry measured by someone moving along it.
This guide develops that distinction from the beginning. We first study a moving point: its velocity, distance travelled and turning. We then study a two-parameter surface: its tangent directions, lengths, areas and bending. The examples lead from lines and circles to helices, cylinders, spheres and saddle-shaped graphs.
Level: undergraduate differential geometry and advanced enrichment. You should already be comfortable with differentiation, integration, vectors, dot products, cross products and matrices. This is not a list of Additional Mathematics examination requirements. For preparation, use the Calculus and Vectors and Coordinate Geometry routes. The BTT Mathematics Hub remains the main entrance.
Reading route: curves and regularity → arc length → curvature and torsion → surface patches → fundamental forms → intrinsic geometry → practice → worked solutions.
The central idea
A parametrised curve γ(t) assigns a position to each parameter value. Its derivative tells us how that position changes. A parametrised surface X(u,v) assigns a position to two parameters. Its two first derivatives describe the directions available for movement along the surface.
First derivatives provide tangent directions and the information needed to measure lengths and areas. Second derivatives reveal bending. However, raw derivative components change when coordinates change. Differential geometry constructs quantities and objects whose meaning remains consistent across those changes.
The standard theory of regular curves and surfaces used here is developed in Weiyi Zhang’s Geometry of Curves and Surfaces. The calculations below are self-contained teaching examples; the references provide fuller proofs and further theory.
1. A parametrisation is a journey, not just a trace
Let I be an interval and let γ:I→R³ be a smooth map. The image γ(I) is its geometric trace. The map also records the order and speed with which the trace is visited. Those are different pieces of information.
For example, γ(t)=(cos t,sin t,0), with 0≤t≤2π, travels around the unit circle once. The map η(t)=(cos 2t,sin 2t,0), over the same interval, travels around it twice. The traces agree, but the journeys do not have the same total distance. A length integral counts traversal, including repeated visits. It does not automatically measure the image only once.
The velocity is γ′(t). The curve is regular at t when γ′(t)≠0. Regularity ensures that a nonzero tangent direction is available. A regular parametrisation may still cross itself at different parameter values; regularity is a local derivative condition, not a guarantee of global injectivity.
This distinction prevents a common mistake. An immersed figure-eight can have nonzero velocity everywhere while two different parameter values represent the crossing point. An embedded curve requires additional control of the map onto its image.
A failed parameter can hide an ordinary line
Take γ(t)=(t³,0,0). Its derivative vanishes at t=0, so this parametrisation is not regular there. Nevertheless, its image is a straight line, which has the perfectly regular parametrisation η(s)=(s,0,0). A zero derivative therefore diagnoses the given representation first. It does not, by itself, prove the underlying image has a geometric singularity.
Contrast this with the cusp γ(t)=(t²,t³,0). Its image near the origin does not form a smooth embedded one-dimensional submanifold. Both examples have zero velocity at zero, but their geometric situations differ. A useful analysis separates the map, its image and the structure being claimed.
2. Speed and arc length
Speed is the magnitude of velocity: v(t)=‖γ′(t)‖. For a continuously differentiable curve on [a,b], the travelled length is L=∫ₐᵇ‖γ′(t)‖dt. The norm is essential: reversing direction does not subtract distance already travelled.
If a curve is regular, define an arc-length coordinate s(t)=∫ₜ₀ᵗ‖γ′(q)‖dq. Then ds/dt is positive. Locally, t can be expressed smoothly in terms of s, and the reparametrised curve has speed one. In arc-length coordinates, one unit of parameter corresponds to one unit of distance along the curve.
This removes an artificial source of variation. A curve should not be judged more sharply bent merely because a point moves along it faster. Arc length supplies the geometric denominator needed to distinguish turning from the speed of traversal.
Worked example: a helix with a known travel distance
Consider γ(t)=(2 cos t,2 sin t,3t). Differentiating gives γ′(t)=(−2 sin t,2 cos t,3). Its squared speed is 4 sin²t+4 cos²t+9=13. Thus the speed is √13, and one full revolution, from t=0 to t=2π, has length 2π√13.
The vertical rise is 6π, while the horizontal projection travels a circle of circumference 4π. The helix length is larger than either of these separate quantities. Its unit-speed parametrisation is γ̃(s)=(2 cos(s/√13),2 sin(s/√13),3s/√13).
The check is direct: differentiating γ̃ produces a vector of norm one. This is stronger evidence than merely recognising a familiar helix formula.
3. Reparametrisation changes the clock
Let t=φ(q), where φ is a smooth change of parameter with smooth inverse and nonzero derivative. The new curve is η(q)=γ(φ(q)). By the chain rule, η′(q)=γ′(φ(q))φ′(q). Therefore its speed is ‖γ′(φ(q))‖|φ′(q)|.
The change-of-variables theorem shows that length is preserved when the same segment is traversed once. A positive φ′ preserves orientation; a negative φ′ reverses it. The absolute value protects length from that reversal.
Not every smooth substitution is an admissible regular reparametrisation. The map q↦q³ has zero derivative at zero, and its inverse is not smooth there. It can create a stationary parameter even when the original curve was regular. Always check the inverse and the derivative before using a reparametrisation theorem.
4. Curvature measures turning per unit distance
The unit tangent is T=γ′/‖γ′‖. For an arc-length parametrised curve, curvature is κ=‖dT/ds‖. It measures the rate at which the direction changes as we move along the curve, not the rate at which a clock advances.
Because T·T=1, differentiation gives T·T′=0. The derivative of a unit tangent is perpendicular to the tangent itself. Turning therefore occurs in a normal direction.
For a regular C² space curve with an arbitrary parameter, the convenient formula is κ=‖γ′×γ″‖/‖γ′‖³. For a plane curve (x(t),y(t)), it becomes κ=|x′y″−y′x″|/(x′²+y′²)³ᐟ². A signed plane curvature can omit the absolute value once an orientation of the plane is fixed. See the curvature development in Zhang’s curve notes.
When coordinates have units of length, curvature has units of inverse length. A small circle bends more sharply than a large circle. This dimensional check immediately rules out a proposed circle-curvature formula proportional to the radius.
Worked example: the circle and the parabola
For γ(t)=(R cos t,R sin t), with R>0, the numerator |x′y″−y′x″| equals R², while the denominator is R³. Hence κ=1/R. The value does not depend on t because every part of the circle has the same bending.
For γ(t)=(t,t²/2), the first derivative is (1,t) and the second is (0,1). Thus κ(t)=1/(1+t²)³ᐟ². At the vertex, κ=1. Far from the vertex, κ becomes small. The graph has increasing slope, yet decreasing curvature away from the vertex. Steepness and bending are not the same measurement.
A straight line with a nonconstant speed is another useful test. Its acceleration may be nonzero, but its unit tangent remains constant, so its curvature is zero. Acceleration contains both speed change and direction change; curvature isolates the latter geometrically.
5. The moving frame and torsion
Where κ>0, define the principal normal N=(dT/ds)/κ and the binormal B=T×N. These three vectors form an orthonormal moving frame. For a sufficiently smooth regular space curve with nonzero curvature, the Frenet–Serret equations are dT/ds=κN, dN/ds=−κT+τB and dB/ds=−τN.
The scalar τ is torsion. It measures the rotation of the osculating plane, the plane spanned by T and N. With the stated right-handed frame convention, τ=det(γ′,γ″,γ‴)/‖γ′×γ″‖². This formula requires the denominator to be nonzero.
For the helix γ(t)=(2 cos t,2 sin t,3t), we obtain γ′×γ″=(6 sin t,−6 cos t,4). Its squared norm is 52. The triple product with γ‴ is 12. Consequently κ=2/13 and τ=3/13.
Neither quantity is the helix’s vertical slope. Curvature describes tangent turning; torsion describes the turning of the osculating plane. The two are independent parts of the local shape of a space curve.
At a straight segment, the usual principal normal is not determined by dT/ds because that derivative is zero. Writing the Frenet frame there without acknowledging this failure hides a genuine hypothesis. Other moving frames can handle situations where the Frenet frame is inconvenient.
6. From one parameter to two
A surface patch is described by a smooth map X:U→R³, where U is open in R². To represent an embedded patch, X must be a homeomorphism onto its image, and the derivatives Xᵤ and Xᵥ must be linearly independent. Equivalently, Xᵤ×Xᵥ≠0.
The derivative condition says that the two parameters create two genuinely independent local directions. If changing u and changing v only move along the same line, the formula has not supplied a regular two-dimensional patch.
For a smooth graph X(u,v)=(u,v,f(u,v)), the partial derivatives are Xᵤ=(1,0,fᵤ) and Xᵥ=(0,1,fᵥ). Their cross product is (−fᵤ,−fᵥ,1), which never vanishes. Every smooth graph therefore gives a regular surface patch over its open parameter domain.
By contrast, a formula can lose rank because of its coordinates rather than because the surface itself is singular. Spherical longitude and colatitude coordinates degenerate at the poles. The sphere remains smooth there; another chart is needed.
7. Tangent planes are first-order approximations
At p=X(u₀,v₀), the tangent space is the vector space spanned by Xᵤ and Xᵥ at that parameter value. The affine tangent plane in R³ is p plus that vector space. The distinction is small in a drawing but important in calculations: tangent vectors are displacements, while points of the affine plane have positions.
A unit normal on the patch is N=(Xᵤ×Xᵥ)/‖Xᵤ×Xᵥ‖, or its negative. Choosing one fixes a local orientation. The tangent plane equation is N·(q−p)=0.
Worked example: the saddle graph z=uv
At (u,v)=(1,2), the surface point is p=(1,2,2). The tangent vectors are Xᵤ=(1,0,2) and Xᵥ=(0,1,1). Their cross product is (−2,−1,1), so an upward unit normal is (−2,−1,1)/√6.
The tangent plane is −2(x−1)−(y−2)+(z−2)=0, or z−2=2(x−1)+(y−2). Substituting p checks that the plane passes through the required point. Dotting the normal with both tangent vectors checks that the plane has the required directions.
These checks catch different mistakes. Passing through the point does not verify the normal, and finding a correct normal does not protect against a misplaced constant in the plane equation.
8. The first fundamental form measures movement
Define E=Xᵤ·Xᵤ, F=Xᵤ·Xᵥ and G=Xᵥ·Xᵥ. The first fundamental form is the quadratic expression I=E du²+2F du dv+G dv². Its matrix is the symmetric matrix with rows (E,F) and (F,G).
Why does it appear? A curve on the surface has the form c(t)=X(u(t),v(t)). By the chain rule, c′=Xᵤu′+Xᵥv′. Taking its squared norm gives ‖c′‖²=E(u′)²+2Fu′v′+G(v′)². The formula is simply the Euclidean dot product expressed in surface coordinates.
The same matrix measures angles and inner products of tangent vectors. It is positive definite because a nonzero parameter velocity produces a nonzero tangent vector on a regular patch. In particular E>0 and EG−F²>0. The first form and its coordinate-independent interpretation are developed in Nigel Hitchin’s Geometry of Surfaces, Section 4.2.
For the graph z=uv, E=1+v², F=uv and G=1+u². At (1,2), the matrix has rows (5,2) and (2,2). A parameter displacement with components (a,b) has squared tangent length 5a²+4ab+2b². Treating those components as ordinary Cartesian coordinates would give the wrong length.
9. Surface area and the missing scale factor
A small parameter rectangle is carried to a small tangent parallelogram. Its area is scaled by ‖Xᵤ×Xᵥ‖. The surface area element is therefore dA=√(EG−F²) du dv, and the area of a one-to-one patch over a region D is ∫∫ᴰ√(EG−F²) du dv.
The identity ‖a×b‖²=(a·a)(b·b)−(a·b)² explains the determinant. The area formula is not an unrelated rule to memorise; it is the same metric information assembled into a two-dimensional measurement.
For a graph z=f(u,v), the factor is √(1+fᵤ²+fᵥ²). Thus integrating 1 over the parameter domain gives projected area, while integrating this factor gives surface area. The two agree for a horizontal plane but generally differ for a sloping or curved graph.
If the parametrisation covers a region multiple times, the integral counts multiplicity. Checking injectivity or using a suitable division into patches is part of the area calculation, not an optional geometric afterthought.
Worked example: a paraboloid cap
Take f(u,v)=(u²+v²)/2 over the unit disc. Then fᵤ=u and fᵥ=v, so dA=√(1+u²+v²) du dv. Switching to polar coordinates gives A=2π∫₀¹r√(1+r²)dr.
With w=1+r², the integral becomes A=(2π/3)(2√2−1). It exceeds the projected disc area π, as expected. The check is geometric: a graph whose slope is nonzero over a substantial region must have more area than its perpendicular projection onto the parameter plane.
10. Cylinders and spheres: coordinates are not equal rulers
For a cylinder of radius R, use X(u,v)=(R cos u,R sin u,v), restricting u to a suitable open interval for a single chart. Then E=R², F=0 and G=1. A coordinate change of one radian around the cylinder travels distance R; a change of one length unit vertically travels distance one.
Hence I=R²du²+dv² and dA=R du dv. The lateral area over one revolution and height h is 2πRh. The angular seam can be treated by overlapping charts or as a measure-zero seam in this integral. It does not create a physical boundary in the whole cylinder.
For a sphere, write X(θ,φ)=(R sinθ cosφ,R sinθ sinφ,R cosθ), away from the poles and an angular seam. The first form is I=R²dθ²+R²sin²θ dφ². The longitude scale shrinks towards a pole. Equal changes in longitude do not represent equal physical distances at different latitudes.
The area factor is R²sinθ for 0<θ<π. Integrating over the sphere gives 4πR². At θ=0 or π the coordinate expression degenerates; this warns us about the chart, not about the smoothness of the sphere.
11. The second fundamental form measures bending into the normal direction
Choose a smooth unit normal N. Define e=Xᵤᵤ·N, f=Xᵤᵥ·N and g=Xᵥᵥ·N. The second fundamental form is II=e du²+2f du dv+g dv². Unlike the first form, it depends on the chosen normal orientation.
For a surface curve c(t)=X(u(t),v(t)), differentiate twice. The terms containing u″ and v″ are tangent, so they disappear when dotted with N. What remains is N·c″=e(u′)²+2fu′v′+g(v′)². This is the normal component of the curve’s acceleration.
For a nonzero tangent direction with parameter components (a,b), its signed normal curvature is κₙ=II(a,b)/I(a,b). Dividing by I corrects for the size of the tangent vector. A direction should not acquire a different curvature merely because we describe it using twice as large a vector.
The construction follows the second-form framework in Hitchin’s Section 4.4. Sign conventions vary between treatments; this guide uses the shape operator S=−dN, so II(v,w)=⟨Sv,w⟩.
12. Principal, Gaussian and mean curvature
The shape operator is a linear map on the tangent space. In the coordinate basis, its matrix is I⁻¹II, where I and II now denote their matrices. Its two real eigenvalues κ₁ and κ₂ are the principal curvatures. They are the extreme normal curvatures over unit tangent directions.
Gaussian curvature is K=κ₁κ₂=(eg−f²)/(EG−F²). Mean curvature is H=(κ₁+κ₂)/2=(Eg−2Ff+Ge)/(2(EG−F²)). These expressions use the metric as well as the second derivatives. Diagonalising the second-form matrix alone generally does not give the principal curvatures unless the coordinate basis is orthonormal.
Reversing N reverses II and both principal curvatures. Mean curvature changes sign, while Gaussian curvature does not. Therefore two texts can give opposite signs for mean curvature of the same oriented example if their definitions of the shape operator differ. A stated convention resolves the discrepancy.
Worked comparisons: sphere, cylinder and saddle
For a sphere of radius R with outward normal N(p)=p/R, differentiation gives dN(v)=v/R. Therefore S(v)=−v/R. Both principal curvatures are −1/R, giving K=1/R² and H=−1/R under our convention.
For a cylinder with outward normal, the circumferential principal curvature is −1/R and the vertical principal curvature is zero. Hence K=0 and H=−1/(2R). The cylinder bends in the surrounding space while having zero Gaussian curvature.
For X(u,v)=(u,v,(u²−v²)/2), the first form at the origin is the identity matrix. With upward normal, the second-form matrix there is diag(1,−1). Thus K=−1 and H=0. The vanishing mean curvature at this point does not make the surface flat there: its two principal curvatures cancel in the average.
13. Intrinsic geometry is not the same as visible bending
A measurement is intrinsic when it belongs to the surface’s own metric geometry rather than to a particular way of placing the surface in R³. The first fundamental form determines curve lengths, angles and areas. These are the measurements available to an observer constrained to the surface.
On a cylinder, set x=Ru and y=v. Then its first form becomes dx²+dy². A suitably cut cylinder patch can be unrolled isometrically onto a plane region: lengths and angles agree. Its second form changes because its bending in R³ changes, but its intrinsic metric does not.
Gauss’s Theorema Egregium says Gaussian curvature is determined by the first fundamental form and is preserved by local isometries. A sphere has positive K, whereas a plane has K=0, so there is no local length-preserving flattening of a spherical patch onto a plane. See the intrinsic-curvature discussion in Zhang’s surface theory.
This is a mathematical explanation of the opening comparison. The cylinder can bend without acquiring nonzero Gaussian curvature. The sphere cannot be flattened locally while preserving the metric. A picture saying “both are curved” fails to distinguish the relevant quantities.
14. A curve on a surface has two kinds of turning
A unit-speed curve on a surface has an ambient curvature vector dT/ds. Part points normal to the surface; the remainder lies in its tangent plane and is perpendicular to T. These give normal curvature and geodesic curvature respectively.
A geodesic has zero geodesic curvature, meaning it does not turn sideways within the surface. It can still curve strongly in R³. A great circle on a sphere is the standard example: it is a geodesic of the sphere but is not a straight line in the surrounding space.
For a latitude circle at colatitude θ on a sphere, its radius in R³ is R sinθ, so its ambient curvature magnitude is 1/(R sinθ). The magnitude of normal curvature of the sphere is 1/R. Except at the equator, these are different: the latitude has additional sideways turning within the sphere.
This prepares the later study of Riemannian geometry and geodesics. It also warns against using the curvature of a drawn route as though it were automatically the Gaussian curvature of the surface beneath it.
15. A reliable calculation sequence
For a curve, state the parameter interval and smoothness, calculate velocity, check regularity, and only then compute speed, length and curvature. Introduce torsion only where the Frenet construction is defined. Keep units and orientation visible.
For a surface, state the parameter domain, inspect injectivity locally, and check Xᵤ×Xᵥ. Calculate E,F,G before using any surface-length formula. Choose the normal before e,f,g. Check EG−F²>0 and compare the final curvature with a simple geometric expectation.
In applications, these calculations describe a chosen smooth model. A fitted curve through noisy observations is not an exact physical trajectory, and second derivatives can amplify noise. Mathematical correctness of the formulas does not establish the accuracy of the fitted geometry. An honest application also examines sampling, scale and modelling error.
16. Independent practice: twelve questions
Attempt the questions before reading the solutions. For each one, name the object, the relevant regularity condition and the quantity requested. Use the convention S=−dN throughout.
- For γ(t)=(3t,4t), 0≤t≤2, find speed, length, unit tangent and curvature.
- For γ(t)=(2 cos3t,2 sin3t), find speed and curvature. What interval starting at zero describes one revolution?
- Find the curvature of γ(t)=(t,t²) at t=1.
- For γ(t)=(3 cos t,3 sin t,4t), find speed, curvature, torsion and the length of one revolution.
- For z=uv at (u,v)=(1,2), find the upward unit normal, affine tangent plane and first-form matrix.
- For X(u,v)=(u+v,u−v,0), compute E,F,G and the area factor. Explain why the factor is not one even though the image is flat.
- A cylinder has radius 3 and height 5. Find its lateral area, Gaussian curvature and outward-normal mean curvature.
- Find K and H for a sphere of radius 4 with outward normal.
- At the origin of z=(u²−v²)/2, find both fundamental-form matrices, K and H.
- Compute K and H for z=uv at (u,v)=(1,2), using upward normal.
- A latitude on a sphere of radius R has colatitude π/6. Compare its ambient curvature magnitude with the magnitude of the sphere’s normal curvature.
- Find the area of z=(u²+v²)/2 above u²+v²≤1. Explain why the answer should exceed π.
17. Worked solutions and checks
1–2. Separate movement speed from geometric bending
1. The derivative is (3,4), with norm 5. Length is ∫₀²5dt=10. The unit tangent is (3/5,4/5), which is constant, so curvature is zero. The displacement also has length 10 because the route is a straight segment travelled without reversal.
2. The derivative is (−6 sin3t,6 cos3t), so speed is 6. The trace is a circle of radius 2, hence curvature is 1/2. One revolution requires 3t to run from 0 to 2π, giving 0≤t≤2π/3. Its length is 6(2π/3)=4π, the expected circumference. A faster parameter has not changed the circle’s curvature.
3–4. Use the derivative formulas with their hypotheses
3. Here x′=1, y′=2t, x″=0 and y″=2. Thus κ=2/(1+4t²)³ᐟ². At t=1 this is 2/(5√5). The velocity never vanishes because its first component is one.
4. The speed is √(9+16)=5. For a helix (a cos t,a sin t,bt), direct differentiation gives κ=a/(a²+b²) for a>0 and τ=b/(a²+b²). Substituting a=3,b=4 gives κ=3/25 and τ=4/25. One revolution has length 5·2π=10π. The nonzero curvature makes the Frenet frame available throughout this example.
5–6. Build tangent geometry before measuring
5. The tangent vectors are (1,0,2) and (0,1,1). Their cross product is (−2,−1,1), of norm √6. Thus N=(−2,−1,1)/√6. The tangent plane is z−2=2(x−1)+(y−2). Dot products give E=5,F=2,G=2, so the matrix has rows (5,2) and (2,2). Its determinant is 6, consistent with the squared cross-product norm.
6. Xᵤ=(1,1,0) and Xᵥ=(1,−1,0). Therefore E=2,F=0,G=2 and the area factor is √4=2. The coordinates are orthogonal but each coordinate basis vector has length √2. Flatness does not require a chosen parameter grid to have unit scale.
7–9. Keep the normal convention consistent
7. The lateral area is 2π·3·5=30π. The outward-normal principal curvatures are −1/3 and 0. Consequently K=0 and H=−1/6. End caps were not included in the requested lateral area.
8. Both principal curvatures are −1/4. Thus K=1/16 and H=−1/4. Reversing the normal would change H to 1/4 while leaving K unchanged.
9. At the origin the first derivatives of the graphing function vanish, so I is the identity matrix. Its Hessian is diag(1,−1), and the upward normal is (0,0,1), so II=diag(1,−1). Hence K=−1 and H=0. The second form is nonzero despite the vanishing mean.
10–12. Combine the layers
10. From Question 5, E=5,F=2,G=2 and EG−F²=6. The second derivatives are Xᵤᵤ=0, Xᵥᵥ=0 and Xᵤᵥ=(0,0,1), giving e=g=0 and f=1/√6. Therefore K=(−1/6)/6=−1/36. Also H=(−4/√6)/12=−1/(3√6). This calculation shows why the metric determinant cannot be dropped.
11. Since sin(π/6)=1/2, the latitude’s radius is R/2 and its ambient curvature magnitude is 2/R. The magnitude of normal curvature of the sphere is 1/R. The extra turning lies within the tangent plane; this latitude is not a geodesic.
12. The area integral is 2π∫₀¹r√(1+r²)dr=(2π/3)(2√2−1). The factor √(1+r²) is at least one and is strictly larger than one except at the centre. Integrating over a region of positive area therefore gives an answer strictly larger than the projected area π.
18. Questions that reveal whether the ideas are secure
Does zero Gaussian curvature mean the surface is a plane?
No. A cylinder has K=0 while bending in R³. Gaussian curvature measures a particular intrinsic property, not every kind of bending. The entire second fundamental form vanishes on a plane patch, which is a stronger statement.
Can a regular surface have a singular-looking formula?
Yes. A chart may degenerate or stop covering part of the surface. The poles of spherical coordinates are the standard example. Before declaring the surface singular, try another representation and test its rank.
Why are there two fundamental forms?
The first form measures tangent vectors: it answers length, angle and area questions. The second form measures how tangent movement bends into the selected normal direction. They interact, but they answer different questions.
What should be practised before moving on?
Rework one curve and one surface with changed parameters. Show that the resulting length or curvature agrees after the change is interpreted correctly. This transfer task tests whether the formulas are attached to geometric objects rather than memorised coordinate patterns.
Sources and further study
Weiyi Zhang, Geometry of Curves and Surfaces, supplies a full course treatment of regular curves, arc-length parametrisation, moving frames and surface curvature. Nigel Hitchin, Geometry of Surfaces, develops surface charts, fundamental forms, isometries and global surface geometry. Compare definitions before transferring signs or notation between sources.
For the underlying topological distinctions, return to Manifolds, Knots and Geometric Topology. The next differential-geometry step is to define smoothness without requiring one global picture in Euclidean space: smooth manifolds and smooth maps.