MATH F342 (Differential Geometry)
BITS Pilani
Pilani Campus
Dr. Sangita Yadav
Department of Mathematics
BITS Pilani, Pilani Campus, Rajasthan
Course Structure
Instructor-Incharge: Dr. Sangita Yadav
Text Book: Somasundaram, D., Differential Geometry A
First Course, Narosa Publishing House (2005).
Objectives: To study the geometry of curves and
surfaces in 3- dimensional space.
Expected outcomes:
To understand how Calculus is useful in formulating
and further studying the geometric concepts
To set a background for geometric concepts in
higher dimensions and more abstract settings
The glimpse of applications of geometry to map
making etc.
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January 20, 2022
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Books
Reference Books:
1
Pressley, A – Elementary Differential Geometry, 2nd
Edition(Corrected Print), Springer (2012)
2
Gray A, Abbena E, Salamon S – Modern differential
geometry of curves and surfaces with
MATHEMATICA, 3rd Edition, CRC Press (2006)
3
Oprea, J – Differential Geometry and Its
Applications, Mathematical Association of
America(2007)
4
Bär, Christian - Elementary Differential Geometry,
1st South Asian edition, Cambridge University Press
(2011)
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Curves-The Beginning
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The Straight line
Slope and intercept form
y = mx + c.
Slope Point Form
y − y0 = m(x − x0 ).
Two Point Form
1
y − y2 = xy22 −y
−x1 (x − x2 )
provided x1 ̸= x2 .
Intercept Form
y
x
a + b = 1, a ̸= 0, b ̸= 0.
General Form
ax + by + c = 0; (a, b) ̸= (0, 0).
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Parabola
y = x2 ;
Ellipse
x2
a2
+
y2
b2
= 1; , ab ̸= 0.
Hyperbola
Circle
x2 + y 2 = r2 ; r > 0.
Dr. Sangita Yadav (BITS Pilani)
x2
a2
−
y2
b2
= 1; , ab ̸= 0.
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Graph of a continuous Function
y = f (x)
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More Examples
Nodal Cubic:
x3 + x2 − y 2 = 0
Cuspidal: y 2 − x3 = 0
For curves above, tangent at origin does not exist.
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Curves in 3-dimensions
Line: Line is given as intersection of two nonparallel
planesa1 x + b1 y + c1 z + d1 = 0,
a2 x + b2 y + c2 z + d2 = 0.
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Curves in 3-dimensions
More generally, intersection of two surfaces in
3-dimensions colorredmay define a curve in 3-dimensions.
These are common solutions of the system
f (x, y, z) = 0
g(x, y, z) = 0
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More Examples
x2 + y 2 = 1,
C :
x + y + z = 0.
It gives an ellipse in 3dimension.
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x2 + y 2 + z 2 = 1,
C :
x + y + z = 0.
Its a great circle of the
sphere.
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C : x2 + y 2 + z 2 = 1, z = 1: Not a curve! Single
point.
C : x2 + y 2 + z 2 = 1, z = 2: Not a curve! Empty
set.
C : x2 + y 2 + z 2 = 1, 2x2 + 2y 2 + 2z 2 = 2: Not a
curve! Sphere.
C : x2 + y 2 + z 2 = 1, z = 3/4: A circle.
If we get a curve in this manner, the representation is
called a level curve (in plane for one equation in 2
variables, and in 3 dimensions for 2 equations in 3
variables). In this representation, given a point, it is easy
to verify if it lies on the curve.
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Parametric representation
Coordinatess x, y, z of the point on the curve are given
as a function of a single parameter t.
Thus (x(t), y(t), z(t)); t ∈ I for an interval I gives a
parametric represntation of a curve in 3 dimensions.
x,y,z are required to be atleast continuous, we impose
further conditions for convinience.
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Examples
Straight line: (a1 t + b1 , a2 t + b2 , a3 t + b3 ) : t ∈ R
where ai , bi are constants and
(a1 , a2 , a3 ) ̸= (0, 0, 0).
Circle: (cos t, sin t) : t ∈ R.
Circular helix: (cos t, sin t, t) : t ∈ R.
Twisted Cubic: (t, t2 , t3 ) : t ∈ R. It is nothing but
intersection of two surfaces y − x2 = 0 and
z − x3 = 0.
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Twisted Cubic
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Relation between two representations
To expand on this example, let us consider how this
relationship can be structured.
Graph of a function: For the function f (x) for a real
variable x ∈ I, its graph is given by
{(x, y) : y = f (x)}.
Its level curve form is F (x, y) = y − f (x) = 0.
Its parametric form is x = t, y = f (t).
One can easily go from one form to another form.
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Extending this further, suppose a curve in 3-dimension is
given by y = f (x), z = g(x).
Its parametric form is x = t, y = f (t), z = g(t).
Its level curve form is given by y − f (x) = 0,
z − g(x) = 0.
Common feature in both of these is that a coordinate
can be chosen as parameter.
In general when can this be done?
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A way Forward
If we can rewrite each of equations in level curve as two
of the variables can be written in terms of the third, then
we can write it in parametric form. Thus we have to
solve the system for two variables in terms of the third
variable. How and when can we do this?
If the given curve is in parametric form, we can write the
parameter in terms of one of the variable i.e. we can
write level curve form.This amounts to solving one of the
three equations x = f (t), y = g(t), z = h(t) (Obtained
from the parametrization) for the parameter t in terms of
the corresponding Cartesian variable. How and when can
we do this?
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Implicit and inverse function theorem
While finding explicitly the solution of the system may
not always be possible, mere existence of the solution
ensures the existence of the other representation from
given one. Many a times this also helps.
It may not always be possible to even assure existence of
the solution, but under some conditions we may be able
to ensure existence of a solution in a neighborhood of a
point.
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Implicit Function Theorem
For one equation in two variables
Suppose f (x, y) is a function with continuous partial
derivatives near P (a, b) which lies on the level curve
∂f
(a, b) ̸= 0.
C : f (x, y) = 0 i.e. f (a, b) = 0. Assume
∂y
Then, there exists a neighbourhood N of (a, b), an open
interval Ia containing a and a function ϕ : Ia → R such
that C ∩ N is exactly the graph of ϕ.
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Implicit Function Theorem
For real valued functions f (x, y, z), g(x, y, z), let
∂(f, g)
=
∂(x, y)
∂f
∂x
∂f
∂y
∂g
∂x
∂g
∂y
∂(f, g)
,
=
∂(y, z)
∂f
∂y
∂f
∂z
∂g
∂y
∂g
∂z
be the 2 × 2 minors of the jacobian matrix
" ∂f ∂f
J(f, g)
∂x
∂y
= ∂g ∂g
J(x, y, z)
∂x
Dr. Sangita Yadav (BITS Pilani)
∂y
MATH F342 (Differential Geometry)
,
∂(f, g)
=
∂(x, z)
∂f
∂x
∂g
∂x
∂f
∂z
∂g
∂z
∂f #
∂z
∂g
∂z
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Implicit Function Theorem
For curves in 3-dimensions
Let f (x, y, z), g(x, y, z) be real valued functions with
continuous first partial derivatives in a neighbourhood of
a common solution (x0 , y0 , z0 ) of f (x, y, z) = 0,
∂(f, g)
g(x, y, z) = 0. If, at (x0 , y0 , z0 ),
̸= 0, then there
∂(y, z)
is a neighbourhood N of (x0 , y0 , z0 ), an open interval I0
containing x0 and real valued functions ϕ, ψ : I0 → R
with continuous derivative such that for the set of
common solutions C of f (x, y, z) = 0, g(x, y, z) = 0 ,
C ∩ N is exactly given by {(x, ϕ(x), ψ(x)) : x ∈ I0 }.
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Remark
Allowing the interchange of the role of variables, we see
J(f,g)
has rank
that if at a point P , the jacobian matrix J(x,y,z)
2, then near the point P , C can be given as a parametric
curve with some coordinate as a parameter.
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Inverse function theorem
Inverse Function Theorem for one variable
Suppose I is an open interval in R and f : I → R be a
function with continuous derivative and x0 ∈ I be such
df
(x0 ) ̸= 0. Then there exists an open interval I0 of
that dx
x0 such that f|I0 has an inverse whose domain is an open
interval J0 and it has continuous derivative on J0 . This
means we can solve the equation f (x) = y for y ∈ J0
with solution in I0 .
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Corollary
Let ⃗γ (t) = (x(t), y(t), z(t)) be a parametric curve with
continuous derivative at interior point t0 of its domain. If
x′ (t0 ) ̸= 0, then there exists a neighbourhood N of
(x(t0 ), y(t0 ), z(t0 )) and an open interval I0 containing t0
and functions f, g : I0 → R with continuous derivative
such that Im(⃗γ ) ∩ N is exactly all points of the level
curve y − f (x) = 0, z − g(x) = 0 in N .
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Geometrical idea of implicit function theorem
Near a point on the curve, curve is approximated by a
tangent line. Near a point of a surface, the surface is
approximated by a tangent plane.
Tangent plane to the level surface f (x, y, z) = 0 at its
point P (x0 , y0 , z0 ) is given by
fx (P )(x − x0 ) + fy (P )(y − y0 ) + fz (P )(z − z0 ) = 0
i.e. ax + by + cz + d = 0 with
(a, b, c) = (fx (P ), fy (P ), fz (P )).
For the intersection of level surfaces, tangent line will be
given by intersection of tangent planes of the level
surfaces.
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Thus tangent line to a level curve f (x, y, z) = 0,
g(x, y, z) = 0 at its point (x0 , y0 , z0 ) is given as solution
of the system
ax + by + cz + d = 0,
lx + my + nz + k = 0
where (a, b, c) = (fx (P ), fy (P ), fz (P )) and
(l, m, n) = (gx (P ), gy (P ), gz (P )). This system can be
solved for y and z in terms of x provided
∂(f, g)
b c
̸= 0.
=
m n
∂(y, z)
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A Principle in Differential Geometry
To obtain a property of a curve or a surface near a point,
first obtain it for tangent there and then explore if we
can transfer it to the curve or surface.
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Other Representation
A point in 3-dimensional space can be given by
cylindrical coordinates (r, θ, z) instead of Cartesian
coordinates (x, y, z). We could consider a representation
where cylindrical coordinates r(t), θ(t), z(t) are given as
functions of parameter t. In this case, we can get the
usual parametrization of Cartesian coordinates
x = r(t) cos(θ(t)), y = r(t) sin(θ(t)), z = z(t).
Similarly, we can give spherical coordinates ρ(t), θ(t),
ϕ(t), as functions of t giving
x = ρ(t) cos(θ(t)) sin(ϕ(t)),
y = ρ(t) cos(θ(t)) sin(ϕ(t)),
z = ρ(t) cos(ϕ(t)).
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For polar curve r = f (θ) in xy-plane, we get the
parametrization
x = f (θ) cos θ, y = f (θ) sin θ
where θ is a parameter.
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Curve as Roulette
Definition
When one shape rolls over the other, a fixed point of the
rolling shape traces a curve called roulette.
Cycloid
A circle rolls over a straight line. A fixed point on the
circle traces a cycloid.
x = a(θ − sin θ), y = a(1 − cos θ)
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Roulettes
Epicycloid
A circle of radius b rolls over a circle of radius a from the
outside.A fixed point on the rolling circle traces the
epicycloid.
θ
x = (a + b) cos θ − b cos (a + b)
b
θ
y = (a + b) sin θ − b sin (a + b)
b
where b is the radius of the rolling
and a that of the fixed circle, and θ
is the angle between the radius vector
of the point of contact of the circles.
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Roulettes
Hypocycloid
A circle of radius b rolls on a circle of radius a from
inside. A fixed point on the rolling circle traces an
hypocycloid.
θ
x = (a − b) cos θ + b cos (a − b)
b
θ
y = (a − b) sin θ − b sin (a − b)
b
where b is the radius of the rolling
and a that of the fixed circle, and θ
is the angle between the radius vector
of the point of contact of the circles.
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Roulettes
Epitrochoid/Hyptrochoid
A disk of radius b rolls on a disk of radius a from
outside/inside. A fixed point on the rolling disk traces an
epitrochoid/hyptrochoid.
θ
x = (a + sb) cos θ + sd cos (a + sb)
,
b
θ
y = (a + sb) sin θ − d sin (a + sb)
,
b
where d is the distance of the point from the center of
rolling circle, s = 1/−1 for epitrochoid/hyptrochoid.
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Locus of points with certain properties
Recall definition of conics
The locus of all points in the plane such that ratio of its
distances from a fixed line(directrix) and a fixed
point(focus) not on the line is constant.
Witch of Agnesi
8a3
Cartesian form: y = 2
x + 4a2
x = 2a tan θ,
Parametric form:
where θ is the
y = 2a cos2 θ
angle between the diameter at origin and line
segment joining the point from the origin.
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Witch of Agnesi
Figure 1: The witch of Agnesi with parameters a = 1, a = 2, a = 4,
and a = 8.
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Thanks for your
attention!
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