lemmathm lemma

# Almost Nonnegative Ricci curvature and new vanishing theorems for genera

Xiaoyang Chen  School of Mathematical Sciences, Institute for Advanced Study,
Tongji University, Shanghai, China<EMAIL_ADDRESS>,  Jian Ge
School of Mathematical Sciences, Laboratory of Mathematics and Complex
Systems, Beijing Normal University, Beijing 100875, P. R. China.
<EMAIL_ADDRESS>and  Fei Han  Department of Mathematics, National University of
Singapore<EMAIL_ADDRESS>
###### Abstract.

We derive several vanishing theorems for genera under almost nonnegative Ricci
curvature and infinite fundamental group, which includes Todd genus,
$\widehat{A}$-genus, elliptic genera and Witten genus. A vanishing theorem of
Euler characteristic number for almost nonnegatively curved Alexandrov spaces
is also proved.

###### Contents

  1. 1 Introduction
  2. 2 Proof of Theorem 1.1, Theorem 1.2 and Theorem 1.3
  3. 3 Vanishing theorems on Euler number
  4. A Some basic facts on elliptic genus and Witten genus

##  1\. Introduction

A classical theorem of Lichnerowicz asserts that a compact spin manifold
carrying a Riemannian metric of positive scalar curvature has vanishing
$\widehat{A}$-genus. In a different direction, Lott conjectured that a compact
spin manifold with _almost nonnegative sectional curvature_ has vanishing
$\widehat{A}$-genus [30]. In this paper we derive several vanishing theorems
for various genera under almost nonnegative Ricci curvature condition. We
start from the Todd genus.

######  Theorem 1.1.

Let $M$ be a compact complex manifold with infinite fundamental group. If $M$
admits a sequence of Kähler metrics with almost nonnegative Ricci curvature,
then the Todd genus of $M$ vanishes.

If a sequence of Riemannian metrics $\\{g_{i}\\}_{n\in\mathbb{N}}$ on a smooth
manifold $M$ satisfies

$\operatorname{Ric}(g_{i})\geq-1\quad{\text{and}}\quad\operatorname{diam}(g_{i})\leq\frac{1}{i},$

for all $i\in\mathbb{N}$, then we say that $\\{g_{i}\\}_{n\in\mathbb{N}}$ have
almost nonnegative Ricci curvature. Here $\operatorname{Ric}(g_{i})$ and
$\operatorname{diam}(g_{i})$ stand for the Ricci curvature and diameter of
$g_{i}$, respectively. If $g$ has nonnegative Ricci curvature, it is clear
that
$g_{i}=\epsilon_{i}g,\epsilon_{i}=\frac{1}{i^{2}\operatorname{diam}(g)^{2}}$
have almost nonnegative Ricci curvature.

Given a compact complex manifold $M^{n}$ of complex dimension $n$, its Todd
genus (or holomorphic Euler number) is defined to be

$\sum_{p=0}^{n}(-1)^{p}\dim H^{0,p}(M,\mathbb{C}),$

where $H^{0,p}(M,\mathbb{C})$ is the Dolbeault cohomology group of $M$. We
emphasize that it is necessary to assume that $M$ has infinite fundamental
group in Theorem 1.1. To see this, note that the Todd genus of complex
projective space is $1$.

Theorem 1.1 can be viewed as a complex analogue of a theorem of Fukaya-
Yamaguchi [15], which says that a compact Riemannian manifold with _almost
nonnegative sectional curvature_ and infinite fundamental group has vanishing
topological Euler number. However, a compact Riemannian manifold with almost
nonnegative Ricci curvature and infinite fundamental group may have nonzero
topological Euler number, see the example constructed by Anderson in [1].

######  Example 1.

Let $X^{n+1}$ be a smooth abelian variety of complex dimension $(n+1)\geq 3$
embedded in some complex projective space $\mathbb{CP}^{N}$. Let $M^{n}$ be
the intersection of $X$ with some generic $\mathbb{CP}^{N-1}$. Then $M^{n}$ is
a smooth hypersurface of $X$ with positive holomorphic normal bundle $L$. By
Lefschetz hyperplane theorem, the fundamental group of $M$ is an infinite
abelian group. Moreover, the Todd genus of $M$ is nonzero. In fact, by
Hirzebruch-Riemann-Roch theorem [2, 21], the Todd genus of $M$ is computed by
$\int_{M}\operatorname{td}(TM)$, where $\operatorname{td}(TM)$ is the Todd
class of the tangent bundle of $M$. Let $d$ be the first Chern class of $L$.
As $X$ is an abelian variety, then $\operatorname{td}(TX)=1$. Moreover, by
properties of Todd class [21], we have

$\operatorname{td}(TM)\operatorname{td}(L)=\operatorname{td}(TX)|_{M},$
$\operatorname{td}(L)=\frac{d}{1-e^{-d}},\ d=c_{1}(L).$

Then

$\operatorname{td}(TM)=\frac{1-e^{-d}}{d}.$

As $L$ is a positive line bundle, we see that

$\int_{M}\operatorname{td}(TM)=(-1)^{n}\int_{M}\frac{d^{n}}{(n+1)!}\neq 0.$

Hence the Todd genus of $M$ is nonzero. It follows that
$M\times\mathbb{CP}^{k}$ has nonzero Todd genus for any $k\geq 1$. Therefore
by Theorem 1.1, $M\times\mathbb{CP}^{k}$ does not admit a sequence of Kähler
metrics with almost nonnegative Ricci curvature. For $k\geq 2$, it seems that
all previous known obstructions to almost nonnegative Ricci curvature do not
apply to $M\times\mathbb{CP}^{k}$.

Riemannian manifolds with almost nonnegative Ricci curvature have been studied
extensively [4, 14, 17, 24, 25, 26, 44]. We briefly recall some previously
known results (here $m=\dim M$).

  * •

The fundamental group of $M$ has a nilpotent subgroup of finite index [26].

  * •

The first Betti number of $M$ is bounded above by $m$ [17, 16] with equality
being achieved if and only if $M$ is diffeomorphic to a torus [8, 12].

  * •

If $M$ is spin and $m$ is divisible by $4$, then its $\widehat{A}$-genus is
bounded from above by $2^{\frac{m}{2}}$ [17, 16].

Compact Kähler manifolds with almost nonnegative Ricci curvature have been
studied in [14, 44]. Under the additional assumptions that $M$ has bounded
sectional curvature and other things, it was shown in [44] that $M$ has finite
fundamental group. Moreover, $M^{n}$ is diffeomorphic to a complex manifold
$X$ such that the universal covering of $X$ has a decomposition:
$\widetilde{X}=X_{1}\times\cdots\times X_{s}$, where $X_{i}$ is a Calabi-Yau
manifold, or a hyperKähler manifold, or $X_{i}$ satisfies
$H^{p,0}(X_{i},\mathbb{C})=0,p>0.$ If $M$ has infinite fundamental group,
under the stronger assumption that $M$ has almost nonnegative bisectional
curvature and bounded sectional curvature, it was shown in [14] that there is
a holomorphic fibration $M\rightarrow J(M)$, where $J(M)$ is the Jacobian of
$M$, a complex torus of dimension $\frac{1}{2}b_{1}(M)$ and $b_{1}$ is the
first Betti number of $M$.

Compared with the works in [14, 44], we do _not_ assume boundedness of
sectional curvature in Theorem 1.1.

Our method can also be used to prove the following vanishing result for
$\widehat{A}$-genus.

######  Theorem 1.2.

Let $M$ be a $4m$-dimensional compact spin manifold with infinite fundamental
group. If $M$ admits a sequence of Riemannian metrics with almost nonnegative
Ricci curvature, then its $\widehat{A}$-genus vanishes.

Lott [30] conjectured that a compact spin manifold $M$ with _almost
nonnegative sectional curvature_ has vanishing $\widehat{A}$-genus. Theorem
1.2 weakens the assumption from almost nonnegative sectional curvature to
almost nonnegative Ricci curvature under the additional assumption that $M$
has infinite fundamental group. We emphasize that this additional assumption
is necessary. In fact, the $K3$ surface is a simply connected spin manifold
which admits a Ricci flat metric. However, its $\widehat{A}$-genus is nonzero.

As an application of Theorem 1.2, we give the following example, where all
previous known obstructions to almost nonnegative Ricci curvature do not
apply.

######  Example 2.

Let $B^{8}$ be a Bott manifold, which is spin, simply connected and
$\widehat{A}(B^{8})=1$ (c.f. Section 4 in [37]). Let
$M^{8}=(\mathbb{T}^{2}\times S^{6})\sharp B^{8}$. Then $M^{8}$ is a spin
manifold with infinite fundamental group and

$\widehat{A}(M^{8})=\widehat{A}(\mathbb{T}^{2}\times
S^{6})+\widehat{A}(B^{8})=1.$

By Theorem 1.2, $M^{8}$ can not admit a sequence of Riemannian metrics with
almost nonnegative Ricci curvature.

The elliptic genera and Witten genus are $q$-deformations of the
$\widehat{A}$-genus and $\widehat{L}$-genus. They are partition functions of
two-dimensional quantum field theories and formally the indices of Dirac
operator in free loop spaces. If the formal Chern roots of the complexified
tangent bundle $TM\otimes\mathbb{C}$ of a $4m$-dimensional compact oriented
manifold $M$ are $\\{\pm 2\pi\sqrt{-1}x_{j},1\leq j\leq 2m\\}$, then the
Hirzebruch $\widehat{A}$-genus and $\widehat{L}$-genus are the characteristic
number of $M$ defined by

$\widehat{A}(M)=\left\langle\prod_{j=1}^{2m}\frac{\pi\sqrt{-1}x_{j}}{\sinh(\pi\sqrt{-1}x_{j})},[M]\right\rangle,$
$\widehat{L}(M)=\left\langle\prod_{j=1}^{2m}\frac{2\pi\sqrt{-1}x_{j}}{\tanh(\pi\sqrt{-1}x_{j})},[M]\right\rangle,$

while the elliptic genera $Ell_{1}(M),Ell_{2}(M)$ and the Witten genus $W(M)$
are defined by

$Ell_{1}(M)=\left\langle\left(\prod_{j=1}^{2m}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{1}(x_{j},\tau)}{\theta_{1}(0,\tau)}\right)\\!,[M]\right\rangle=\widehat{L}(M)+O(q),$
(1.1)
$Ell_{2}(M)=\left\langle\left(\prod_{j=1}^{2m}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{2}(x_{j},\tau)}{\theta_{2}(0,\tau)}\right)\\!,[M]\right\rangle=\widehat{A}(M)+O(q^{1/2}),$
(1.2)
$W(M)=\left\langle\left(\prod_{j=1}^{2m}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\right)\\!,[M]\right\rangle=\widehat{A}(M)+O(q^{1/2}),$
(1.3)

where $\theta,\theta_{1},\theta_{2}$ are the classical Jacobi theta functions.
The theory of elliptic genera and Witten genus lead to the profound theory of
elliptic cohomology [22]. See Appendix A for some basic facts on the elliptic
genera and Witten genus.

We can also prove a vanishing theorem for elliptic genera and Witten genus. In
the following, $R_{g_{i}}$ and $V_{i}$ stand for the Riemannian curvature
tensor and volume of $g_{i}$ respectively.

######  Theorem 1.3.

Let $M$ be a $4m$-dimensional compact manifold with infinite fundamental
group. If for some constants $p>2m$ and $\Lambda>0$, there exists a sequence
of Riemannian metrics $\\{g_{i}\\}_{i\in\mathbb{N}}$ on $M$ such that the
following holds for all $i$,

$\operatorname{Ric}(g_{i})\geq-1,\quad\operatorname{diam}(g_{i})\leq\frac{1}{i},$
$\frac{1}{V_{i}}\int_{M}|R_{g_{i}}|^{p}dV_{i}\leq\Lambda,$

then

  1. (i)

$M$ has vanishing elliptic genera if $M$ is spin. In particular, $M$ has
vanishing signature.

  2. (ii)

$M$ has vanishing Witten genus if $M$ is further string (i.e. spin and the
spin class $\frac{p_{1}(M)}{2}=0$).

The integral curvature assumption in Theorem 1.3 is technical. We do not know
if it is indeed necessary.

######  Example 3.

(i) Let $N^{8}=(\mathbb{T}^{2}\times S^{6})\sharp\mathbb{H}P^{2}$, where
$\mathbb{H}P^{2}$ is the quarternionic projective plane. Then $N^{8}$ is a
spin manifold with infinite fundamental group and the signature

$\sigma(N^{8})=\sigma(\mathbb{T}^{2}\times
S^{6})+\sigma(\mathbb{H}P^{2})=1\neq 0.$

So $N^{8}$ can not admit a sequence of Riemannian metrics with almost
nonnegative Ricci curvature as well as the additional integral curvature bound
in Theorem 1.3. Note that the $\widehat{A}$-genus of $N^{8}$ is 0.

(ii) Let $W^{24}$ be a $24$-dimensional simply connected string manifold with
nonzero Witten genus and vanishing $\widehat{A}$-genus. Let
$M^{24}=(\mathbb{T}^{2}\times S^{22})\sharp W^{24}$. Then the Witten genus

$\phi_{W}(M^{24})=\phi_{W}(\mathbb{T}^{2}\times
S^{22})+\phi_{W}(W^{24})=\phi_{W}(W^{24})\neq 0.$

So $M^{24}$ can not admit a sequence of Riemannian metrics with almost
nonnegative Ricci curvature and the additional integral curvature bound in
Theorem 1.3. Note that $M^{24}$ admits a Riemannian metric with positive
scalar curvature.

Some special cases of Theorem 1.1, Theorem 1.2 and Theorem 1.3 are easy to
prove: If $M$ has infinite fundamental group and admits a Riemannian metric
with nonnegative Ricci curvature, then by the Splitting Theorem of [9], some
finite cover of $M$ is diffeomorphic to a product manifold
$\mathbb{T}^{k}\times N$ for some $k\geq 1$. It follows that the genera of $M$
appearing in Theorem 1.1, Theorem 1.2 and Theorem 1.3 are zero.

On the other hand, based on a fibration theorem proved in [25], a vanishing
theorem on signature is proved in [6] for Riemannian manifolds with _almost
nonnegative sectional curvature_ and infinite fundamental group. However, no
such fibration theorem exists under almost nonnegative Ricci curvature
condition, by the counterexample constructed in [1]. In order to prove our
vanishing results on genera, we employ an analytic method based on the Bochner
technique. Via the Atiyah-Singer index theorem, our strategy is to show that
the indices of certain Dirac operators are zero. Using the work of Bérard [4],
we can derive a bound of the indices of those Dirac operators. In the presence
of infinite fundamental group, our crucial contribution is to show that the
indices are in fact zero. In this step the multiplicity property of genera
under a finite covering is used in an essential way. See section 2 for the
details of proof.

In [10], we also obtained several vanishing theorems for genera under
different curvature assumptions. We emphasize that the methods used in the
proof in [10] and this paper are different.

Acknowledgements Xiaoyang Chen is partially supported by National Natural
Science Foundation of China No.11701427. He thanks Professor Botong Wang for
helpful discussions. Fei Han is partially supported by the grant AcRF
A-8000451-00-00 from National University of Singapore. Jian Ge is partially
supported by National Key R&D Program of China grant 2020YFA0712800, NSFC
11731001.

##  2\. Proof of Theorem 1.1, Theorem 1.2 and Theorem 1.3

In this section we prove Theorem 1.1, Theorem 1.2 and Theorem 1.3.

First we prove Theorem 1.1. Let $M$ be a compact complex manifold of complex
dimension $n$ with infinite fundamental group. We are going to show that the
Todd genus of $M$ vanishes if $M$ admits a sequence of Kähler metrics $g_{i}$
satisfying

$\operatorname{Ric}(g_{i})\geq-1,\quad\operatorname{diam}(g_{i})\leq\frac{1}{i}.$

Then the fundamental group of $M$ contains a nilpotent subgroup of finite
index [26]. Let $M_{1}$ be a finite cover of $M$ such that $\pi_{1}(M_{1})$ is
nilpotent. Then we have the following algebraic lemma where the assumption
that $M$ has infinite fundamental group will be used.

######  Lemma .

There exists a sequence of normal subgroups of finite index

$\pi_{1}(M_{1})=G_{1}\supset G_{2}\supset G_{3}\supset\cdots\supset
G_{j}\supset\cdots$

such that $\lim_{j\rightarrow+\infty}[G_{1}:G_{j}]=\infty$.

###### Proof.

If $\pi_{1}(M_{1})$ is free abelian, i.e. $\pi_{1}(M_{1})\simeq\mathbb{Z}^{k}$
for some $k\geq 1$, we can simply take $G_{j}=(2^{j-1}\mathbb{Z})^{k},j\geq
1$. In general, we will apply a theorem of Hirsch [20]. Since $\pi_{1}(M_{1})$
is a finitely generated nilpotent group, by Hirsch’s theorem [20], it is
residually finite, i.e. the intersection of all normal subgroups of finite
index is the unit element. Let $G_{j}$ be the intersection of all normal
subgroups of $\pi_{1}(M_{1})$ with index $\leq j$, $j=1,2,\ldots.$ By a
theorem of Hall [18], in every finitely generated group, the number of
subgroups of given finite index is finite. Then we see that $G_{j}$ is a
normal subgroup of $\pi_{1}(M_{1})$ with finite index. From the definition of
$G_{j}$, we have

$\pi_{1}(M_{1})=G_{1}\supset G_{2}\supset G_{3}\supset\cdots\supset
G_{j}\supset\cdots,$ $\bigcap_{j}G_{j}=\\{{1}\\}.$

Moreover, we have

$\lim_{j\rightarrow+\infty}[G_{1}:G_{j}]=\infty.$

Otherwise, there exists some constant $C>0$ such that $[G_{1}:G_{j}]\leq C$
for all $j$. As $[G_{1}:G_{j}]$ is a monotone increasing sequence taking
values in $\mathbb{Z}$, there must exist some $j_{0}$ such that
$[G_{1}:G_{j}]=[G_{1}:G_{j_{0}}]$ for all $j\geq j_{0}$. It turns out that
$G_{j}=G_{j_{0}}$ for each $j\geq j_{0}$. Hence $\bigcap_{j}G_{j}=G_{j_{0}}.$
Since $\bigcap_{j}G_{j}=\\{1\\}$, then $G_{j_{0}}=\\{1\\}.$ As
$[G_{1}:G_{j_{0}}]<\infty$, it follows that $\pi_{1}(M_{1})=G_{1}$ is a finite
group, which contradicts that $M$ has infinite fundamental group. ∎

The Dolbeault cohomology group of a compact Kähler manifold $M$ satisfy $\dim
H^{0,p}(M,\mathbb{C})=\dim H^{p,0}(M,\mathbb{C})$ for any $p$. Let

$D_{i}={\partial}+{\partial}^{*}:\oplus_{k}\wedge^{2k,0}(M)\rightarrow\oplus_{k}\wedge^{2k+1,0}(M)$

be the first order elliptic operators on $M$ determined by $g_{i}$. Then the
Todd genus of $M$ is equal to the index of $D_{i}$ for any $i$ by Hodge
theory. Denote $\widehat{g_{i}}^{j},\widehat{D_{i}}^{j}$ be the pulling backs
of $g_{i},D_{i}$ to $M_{j}$, where $M_{j}$ is the finite covering of $M_{1}$
with $\pi_{1}(M_{j})=G_{j}$. Then we can compare the diameter of
$\widehat{g_{i}^{j}}$ with that of $\widehat{g_{i}^{1}}$:

######  Lemma  (Ivanov [23]).

$\operatorname{diam}(\widehat{g_{i}^{j}})\leq\ [G_{1}:G_{j}]\
\operatorname{diam}(\widehat{g_{i}^{1}}).$

###### Proof.

We include the proof given by Sergei Ivanov [23] on Mathoverflow for
completeness. Denote $[G_{1}:G_{j}]=N$. Suppose the contrary, let
$\gamma^{j}:[0,\ell]\to M^{j}$ be a geodesic realizes the diameter of
$(M^{j},\widehat{g_{i}^{j}})$ with
$\ell>N\operatorname{diam}(\widehat{g_{i}^{1}})$. Let $\gamma$ be the image of
$\gamma^{j}$ in $M_{1}$ under the canonical projection. We broke $\gamma$ into
$N$ shorter geodesics of equal length $\ell/N$. Namely

$\gamma:=a_{1}a_{2}\cdots a_{N}.$

Since $\ell/N>\operatorname{diam}(\widehat{g_{i}^{1}})$, there must exists a
strictly shorter geodesic $b_{i}$ connecting the two ends of $a_{i}$ for each
$i\in\\{1,\cdots,N\\}$. Set $\bar{p}=\gamma^{j}(0)$, $p=\gamma(0)$ and
$\bar{q}=\gamma^{j}(\ell)$, $q=\gamma(\ell)$. Let $H$ be the subgroup of
$\pi_{1}(M_{1},p)$ whose elements(loops) lift to closed loops in $M_{j}$.
Clearly $[\pi_{1}(M_{1},p),H]=N$. Now consider the following $(N+1)$ loops
based at $p$:

$s_{0}=e,s_{1}=b_{1}a^{-1}_{1},s_{2}=b_{1}b_{2}(a_{1}a_{2})^{-1},\cdots,s_{N}=b_{1}\cdots
b_{N}(a_{1}\cdots a_{N})^{-1}.$

There must exist two element $s_{k}$ and $s_{r}$ with $k<r$ such that
$(s_{k})^{-1}s_{r}\in H$. Now the path $(s_{k})^{-1}s_{r}\gamma$ still lifts
$p$ and $q$ to $\bar{p}$ and $\bar{q}$, yet it is strictly shorter than
$\gamma^{j}$, a contradiction. ∎

Now we derive a bound of the index of $\widehat{D_{i}}^{j}$.

######  Lemma .

There exists a positive constant $C(n)$ depending only on the dimension $n$
such that for each $i,j$, we have

$|\operatorname{ind}(\widehat{D_{i}}^{j})|\leq C(n).$

######  Remark 2.1.

It is essential for us that the constant $C(n)$ is independent of $i,j$.

###### Proof.

We will prove section 2 using the method in [4]. It is based on the following
lemmas.

######  Lemma .

For a smooth section $s$ of an Euclidean vector bundle $E$ over a Riemannian
manifold $(M,g)$, let

$\|s\|_{\infty}=\sup\\{{|s|_{x}:x\in M}\\},$
$\|s\|_{2}^{2}=\int_{M}\|s\|_{x}^{2}dV_{g},$
$L(s)=\frac{V_{g}\|s\|_{\infty}^{2}}{\|s\|_{2}^{2}},$

where $V_{g}$ is the volume of $(M,g)$. Given a finite dimensional subspace
$F$ of $C^{\infty}(E)$, we have

$\dim F\leq l\sup\\{{L(s):s\in F-0}\\},$

where $l$ is the rank of $E$.

We refer the readers to [4] for the proof of section 2.

######  Lemma .

Let $(M,g)$ be a closed $m$-dimensional smooth Riemannian manifold such that
for some constant $b>0$,

$r_{\min}(g)(\operatorname{diam}(g))^{2}\geq-(m-1)b^{2}.$

If $f\in W^{1,2}(M)$ is a nonnegative continuous functions such that $f\Delta
f\geq-hf^{2}$ (here $\Delta$ is a negative operator) in the sense of
distribution for some nonnegative continuous function $h$ then

$\max_{x\in M}|f|^{2}(x)\leq C(m,p,R,\Lambda)\frac{\int_{M}f^{2}dV}{V(g)},$

where $C(m,p,R,\Lambda)$ is some constant depending only on
$m,p,R=\frac{\operatorname{diam}(g)}{bC(b)}$ and

$\Lambda=\frac{\int_{M}h^{p}dV}{V(g)},p>\frac{m}{2}.$

We refer the readers to see [10] or [4] for the proof of Lemma 2. Several
notations in the above lemma need to be clarified here:

$(1)\ r_{\min}(g)=\inf\\{{\operatorname{Ric}(g)(u,u):u\in TM,g(u,u)=1}\\}$

(2) $V(g)$ is the volume of $(M,g)$.

(3) $C(b)$ is the unique positive root of the equation
$x\int_{0}^{b}(cht+xsht)^{m-1}dt=\int_{0}^{\pi}\sin^{m-1}tdt.$

The following explicit expression of the constant $C(m,p,R,\Lambda)$ derived
in [10] is important for us. Let $v=\frac{m}{2}$ if $m>2$ and $1<v<p$ be
arbitrary for $m=2$. Let $\mu$ be the conjugate of $v$ such that

$\frac{1}{v}+\frac{1}{\mu}=1.$

Then

$C(m,p,R,\Lambda)=\mu^{2K_{1}\frac{p(\mu-1)}{\mu(p-1)-p}}B^{2K_{2}}$ (2.1)

$B=C(m,p)\Lambda^{\frac{1}{2}\frac{\mu-1}{\mu(p-1)-p}}R^{\frac{p(\mu-1)}{\mu(p-1)-p}}+2$
(2.2)

$K_{1}=\sum i{\mu}^{-i},\ K_{2}=\sum{\mu}^{-i},$ (2.3)

where $C(m,p)$ is some positive constant depending only on $m,p$.

Now we apply section 2 and section 2 to prove section 2, namely,

$|\operatorname{ind}(\widehat{D_{i}}^{j})|\leq C(n),\ n=\dim_{\mathbb{C}}M.$

Let $E_{j}=\oplus_{k}\wedge^{2k,0}(M_{j}),F_{i,j}=\ker(\widehat{D_{i}}^{j}).$
Then for any $s\in\ker(\widehat{D_{i}}^{j})$, applying the Bochner formula to
$s$, we get

$\frac{1}{2}\Delta|s|^{2}=|\nabla s|^{2}+\langle\mathcal{F}s,s\rangle.$

As $g_{i}$ is a Kähler metric for each $i$, it is crucial here that the
curvature term $\langle\mathcal{F}s,s\rangle$ is controlled by the Ricci
curvature, see for example Page 82 in [2]. By assumption, we have

$\operatorname{Ric}(g_{i})\geq-1,\ \operatorname{diam}(g_{i})\leq\frac{1}{i}.$

Then $\operatorname{Ric}(\widehat{g_{i}}^{j})\geq-1$ and there exists some
positive constant $C(n)$ depending only on $n$ such that

$\frac{1}{2}\Delta|s|^{2}\geq|\nabla s|^{2}-C(n)|s|^{2}.$

By Kato’s inequality [4], we have

$|\nabla s|\geq|\nabla|s||.$

Then we get

$|s|\Delta|s|\geq-C(n)|s|^{2}.$

Let $G_{j}=\pi_{1}(M_{j}),G_{0}=\pi_{1}(M)$. By section 2, we get

$\operatorname{diam}(\widehat{g_{i}}^{j})\leq[G_{1}:G_{j}]\operatorname{diam}(\widehat{g_{i}}^{1})\leq[G_{1}:G_{j}]\
[G_{0}:G_{1}]\operatorname{diam}(g_{i}).$

As $\operatorname{diam}(g_{i})\leq\frac{1}{i}$, we get

$\operatorname{diam}(\widehat{g_{i}}^{j})\leq\frac{[G_{1}:G_{j}]\
[G_{0}:G_{1}]}{i}.$

Applying section 2 to $|s|$ for any $p>n$ (for example take $p=n+1$), we get

$|s|^{2}_{\infty}=:\max_{x\in M}|s|^{2}(x)\leq
C(n,i,j)\frac{\int_{M}|s|^{2}dV_{i}}{V(g_{i})},$

where $C(n,i,j)$ is some positive constant depending on $n,i,j$ (which can be
described explicitly via Equation 2.1).

Applying section 2 to $F_{i,j}$, we get

$\dim(\ker(\widehat{D_{i}}^{j}))\leq C(n,i,j).$

As $\widehat{D_{i}}^{j}$ is self-adjoint, we can also show that

$\dim(\operatorname{coker}(\widehat{D_{i}}^{j}))\leq C^{\prime}(n,i,j),$

where $C^{\prime}(n,i,j)$ is some positive constant depending on $n,i,j$
(which can be described explicitly via Equation 2.1).

Then

$|\operatorname{ind}(\widehat{D_{i}}^{j})|\leq\max(C(n,i,j),C^{\prime}(n,i,j)).$

For a fixed $j$, we have that $\operatorname{ind}(\widehat{D_{i}}^{j})$ is
independent of $i$. Since $\operatorname{Ric}(\widehat{g_{i}}^{j})\geq-1$ and
$\lim_{i\rightarrow\infty}\operatorname{diam}(\widehat{g_{i}}^{j})=0$ for a
fixed $j$, then we get

$|\operatorname{ind}(\widehat{D_{i}}^{j})|=\lim_{i\rightarrow\infty}|\operatorname{ind}(\widehat{D_{i}}^{j})|\leq
C(n),$

where $C(n)$ is some positive constant depending only on $n$. It is essential
for us that the constant $C(n)$ is independent of $i,j$. ∎

Now we are going to finish the proof of Theorem 1.1. As Todd genus is
multiplicative under a finite covering [21], we have

$|\operatorname{ind}(\widehat{D_{i}}^{1})|\
[G_{1}:G_{j}]=|\operatorname{ind}(\widehat{D_{i}}^{j})|.$

By section 2 and section 2, we get

$|\operatorname{ind}(\widehat{D_{i}}^{j})|\leq C(n),\
\lim_{j\rightarrow\infty}[G_{1}:G_{j}]=\infty.$

Hence for each $i$, we must have

$|\operatorname{ind}(\widehat{D_{i}}^{1})|=0.$

It follows that $M_{1}$ has vanishing Todd genus, so does $M$.

The proof of Theorem 1.2 is similar to that of Theorem 1.1. Here we look at
the Atiyah-Singer spin Dirac operator $D$. The curvature term of the Bochner
formula applied to the kernel of $D$ is bounded from below by the scalar
curvature. Then we can apply section 2 to get a bound of $\hat{A}$-genus. This
in fact was already proved in [17, 16]. In the presence of infinite
fundamental group, then our arguments given above can be used to show that the
$\hat{A}$-genus is in fact zero.

Now we deal with the proof of Theorem 1.3. By Appendix A, $Ell_{2}(M)$ is
determined by $\operatorname{ind}(D\otimes B_{k}(T_{\mathbb{C}}M)),0\leq
k\leq\frac{m}{2}$, where $B_{k}(T_{\mathbb{C}}M)$ involves linear combinations
of tensor product of $TM$ at most to power ${[\frac{m}{2}]}$. The kernel of
$D\otimes B_{j}(T_{\mathbb{C}}M)$ is the space of twisted harmonic spinors.
Then the curvature term of the Bochner formula applied to twisted harmonic
spinors is bounded from below by $-C(m)|R_{g_{i}}|$, where $C(m)$ is some
positive constant depending only on $m$. The rest part of proof is almost
identical to the proof of Theorem 1.1. As we do not assume a pointwise bound
of the Riemannian curvature tensor, in order to get a bound of the dimension
of the space of twisted harmonic spinors, the integral curvature assumption
$\frac{1}{V_{i}}\int_{M}|R_{g_{i}}|^{p}dV_{i}\leq\Lambda$ will be used. In the
presence of infinite fundamental group, then our arguments can be used to show
that $Ell_{2}(M)=0$. By Appendix A,

$Ell_{1}(M,-1/\tau)={(2\tau)}^{2m}Ell_{2}(M,\tau)=0$

In particular, we get that the first term in the $q$-expansion of
$Ell_{1}(M)$: the signature $\sigma(M)=0$. This proves Part (i), concerning
the vanishing of elliptic genera in Theorem 1.3.

Concerning the Witten genus, when $M$ is string, the Witten genus $W(M)$ is
determined by finitely many (depending only on the dimension of $M$) $D\otimes
W_{j}(T_{\mathbb{C}}M)$ by Appendix A, the rest of the proof is identical to
the one for elliptic genera.

##  3\. Vanishing theorems on Euler number

The Euler number of a manifold is not a genus. However, certain vanishing
results about Euler number have been obtained under _almost nonnegative
sectional curvature_ assumption. Recall that a smooth Riemannian manifold $M$
has almost nonnegative sectional curvature if it admits a sequence of
Riemannian metrics $\\{g_{i}\\}_{i\in\mathbb{N}}$ such that

$\operatorname{sec}(g_{i})\geq-1,\quad\operatorname{diam}(g_{i})\leq\frac{1}{i},$

where $\operatorname{sec}(g_{i})$ denotes the sectional curvatures of $g_{i}$.

Fukaya-Yamaguchi proved that a closed manifold with almost nonnegative
sectional curvature and infinite fundamental group has vanishing Euler number.
Their proof is based on a fibration theorem [15] characterizing these
manifolds. We are going to give a new proof of this result based on the idea
developed in previous proofs. In fact our proof works for a larger class of
spaces: almost nonnegatively curved Alexandrov spaces. Alexandrov spaces are
generalizations of Riemannian manifolds with lower sectional curvature bound.
They could have topological singularities, for example the spherical cone over
$\mathbb{CP}^{n}$ with Fubini-Study metric is an Alexandrov space. For
background of Alexandrov spaces, cf [5, 3]. Note that due to the possible
singularities on Alexandrov spaces, there is no fibration theorem available
for almost nonnegatively curved Alexandrov spaces. Yet, we still have:

######  Theorem 3.1.

Let $X$ be an $m$-dimensional almost nonnegatively curved Alexandrov space
with infinite fundamental group, then the Euler number of $X$ vanishes.

###### Proof.

By [42], the fundamental group of $X$ contains a nilpotent subgroup of finite
index. By passing to a finite cover, we can assume that $\pi_{1}(X)$ is
nilpotent. By section 2, there is a sequence of subgroups $G_{j}$ of
$\pi_{1}(X)$ with finite index such that
$\lim_{j\rightarrow+\infty}[G_{1}:G_{j}]=\infty$. Then $\widetilde{X}/G_{j}$
is a finite cover of $X$, where $\widetilde{X}$ is the universal covering of
$X$. Clearly, $\widetilde{X}/G_{j}$ is almost nonnegatively curved by pulling
back the metrics. By the extension of Gromov’s Betti number estimate to
Alexandrov spaces (see the main Theorem of [32]), we get that for any $j$,

$\sum_{p=0}^{m}b_{p}(\widetilde{X}/G_{j})\leq C(m).$

This in particular implies that the absolute value of Euler number of
$\widetilde{X}/G_{j}$ is bounded above by some constant depending only on $m$.
As $\lim_{j\rightarrow+\infty}[G_{1}:G_{j}]=\infty$, we see that the Euler
number of $X$ is zero by the multiplicity property of Euler number under a
finite covering. ∎

The same idea can also be used to prove a vanishing theorem about $L^{2}$
Betti numbers under almost nonnegative sectional curvature condition:

######  Theorem 3.2.

Let $M$ be a compact manifold (or more generally a compact Alexandrov space)
with almost nonnegative sectional curvature and infinite fundamental group,
then all the $L^{2}$ Betti numbers of its universal covering $\widetilde{M}$
vanish.

We refer the reader to [34] for the precise definition of $L^{2}$-Betti
numbers. We only recall the following two facts on $L^{2}$-Betti numbers. See
pages 37, 51 and 476 in [34] for details.

######  Lemma  ([33]).

Let $M$ be a finite connected CW-complex with fundamental group $G$. Suppose
there exists a sequence of normal subgroups of finite index

$G=G_{1}\supset G_{2}\supset G_{3}\supset\cdots\supset G_{j}\supset\cdots$

such that $\bigcap_{j}G_{j}=\\{{1}\\}$, then the $p$-th $L^{2}$ Betti number
of $\widetilde{M}$ is equal to

$\lim_{j\rightarrow+\infty}\frac{b_{p}(\widetilde{M}/G_{j})}{[G_{1}:G_{j}]}.$

######  Lemma .

The Euler number of $M$ is equal to the alternative sum of $L^{2}$ Betti
numbers of $\widetilde{M}$.

###### Proof of Theorem 3.1.

By passing to a finite cover, we can assume that $\pi_{1}(M)$ is nilpotent. By
section 2, there is a sequence of subgroups $G_{j}$ of $\pi_{1}(M)$ with
finite index such that $\lim_{j\rightarrow+\infty}[G_{1}:G_{j}]=\infty$. Since
the Betti numbers $b_{p}(\widetilde{M}/G_{j})$ is bounded above by some
constant depending only on $\dim M$, then we have

$\lim_{j\rightarrow+\infty}\frac{b_{p}(\widetilde{M}/G_{j})}{[G_{1}:G_{j}]}=0.$

By section 3, we see that the $p$-th $L^{2}$ Betti number of $\widetilde{M}$
vanishes. ∎

######  Remark 3.3.

The above proof also works for Alexandrov spaces. In fact by [35], any compact
Alexandrov space $X$ admits a good cover, it follows that $X$ is homotopic to
the nerve of a good cover of $X$. Then Lück’s section 3 still applies.

######  Remark 3.4.

It follows from Theorem 3.2 and section 3 that the Euler number of $M$
vanishes. Therefore this provides a second proof of Theorem 3.1 without using
fibration theorem for almost nonnegatively curved manifolds.

In general, it is not true that the Euler number of a compact manifold with
almost nonnegative Ricci curvature and infinite fundamental group is zero. See
the example in [1]. However, our method can also be used to prove a vanishing
theorem of Euler number under an additional assumption on curvature operators.

######  Theorem 3.5.

Let $M$ be a compact manifold with infinite fundamental group. If for some
constant $C>0$, there exists a sequence of Riemannian metrics
$(g_{i})_{i\in\mathbb{N}}$ on $M$ with almost nonnegative Ricci curvature such
that the curvature operator of $g_{i}$ is bounded below by
$-C\operatorname{Id}$, then the Euler number of $M$ vanishes.

The proof of Theorem 3.5 is similar to that of Theorem 1.1. Here we look at
the Dirac operator $d+d^{*}$. The kernel of $d+d^{*}$ is the space of harmonic
forms. As the curvature operator of $g_{i}$ is bounded below by
$-C\operatorname{Id}$, then the curvature term of the Bochner formula applied
to harmonic forms is bounded from below by some negative constant depending
only on $C$ and $\dim M$. The rest part of proof is identical to the proof of
Theorem 1.1.

##  Appendix A Some basic facts on elliptic genus and Witten genus

In this section, we recap some basic facts about elliptic genus and Witten
genus. Let $M$ be a $4m$-dimensional compact oriented smooth manifold and
$\\{\pm 2\pi\sqrt{-1}x_{j},1\leq j\leq 2m\\}$ denote the formal Chern roots of
$T_{\mathbb{C}}M$, the complexification of the tangent vector bundle $TM$ of
$M$.

Let

$\widehat{A}(M)=\prod_{j=1}^{2m}\frac{\pi\sqrt{-1}x_{j}}{\sinh(\pi\sqrt{-1}x_{j})},\
\ \
\widehat{L}(M)=\prod_{j=1}^{2m}\frac{2\pi\sqrt{-1}x_{j}}{\tanh(\pi\sqrt{-1}x_{j})}$

be the $\widehat{A}$-class and $\widehat{L}$-class of $M$ respectively.

The elliptic genera $Ell_{1}(M),Ell_{2}(M)$ and Witten genus $W(M)$ are
defined by (c.f. [27])

$Ell_{1}(M)=\left\langle\left(\prod_{j=1}^{2m}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{1}(x_{j},\tau)}{\theta_{1}(0,\tau)}\right)\\!,[M]\right\rangle=\widehat{L}(M)+O(q),$
(A.1)
$Ell_{2}(M)=\left\langle\left(\prod_{j=1}^{2m}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\frac{\theta_{2}(x_{j},\tau)}{\theta_{2}(0,\tau)}\right)\\!,[M]\right\rangle=\widehat{A}(M)+O(q^{1/2}),$
(A.2)
$W(M)=\left\langle\left(\prod_{j=1}^{2m}x_{j}\frac{\theta^{\prime}(0,\tau)}{\theta(x_{j},\tau)}\right)\\!,[M]\right\rangle=\widehat{A}(M)+O(q^{1/2}),$
(A.3)

where $\theta,\theta_{1},\theta_{2}$ are the classical Jacobi theta functions:

$\theta(v,\tau)=2q^{1/8}\sin(\pi
v)\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi\sqrt{-1}v}q^{j})(1-e^{-2\pi\sqrt{-1}v}q^{j})],$
(A.4) $\theta_{1}(v,\tau)=2q^{1/8}\cos(\pi
v)\prod_{j=1}^{\infty}[(1-q^{j})(1+e^{2\pi\sqrt{-1}v}q^{j})(1+e^{-2\pi\sqrt{-1}v}q^{j})],$
(A.5)
$\theta_{2}(v,\tau)=\prod_{j=1}^{\infty}[(1-q^{j})(1-e^{2\pi\sqrt{-1}v}q^{j-1/2})(1-e^{-2\pi\sqrt{-1}v}q^{j-1/2})],$
(A.6)

$q=e^{2\pi\sqrt{-1}\tau},\tau\in\mathbb{H}$.

Let $E$ be a complex vector bundle. For any complex number $t$, let

$\Lambda_{t}(E)={\mathbb{C}}|_{M}+tE+t^{2}\Lambda^{2}(E)+\cdots,\quad
S_{t}(E)={\mathbb{C}}|_{M}+tE+t^{2}S^{2}(E)+\cdots$

denote respectively the total exterior and symmetric powers of $E$, which live
in $K(M)[[t]]$. Denote $\widetilde{E}=E-\mathbb{C}^{\mathrm{rk}E}$ in $K(M)$.
Elliptic genus and Witten genus can be expressed as (c.f. [11, 28])

$Ell_{1}(M)=\left\langle\widehat{L}(TM)\mathrm{ch}\left(\Theta\left(T_{\mathbb{C}}M\right)\otimes\Theta_{1}\left(T_{\mathbb{C}}M\right)\right),[M]\right\rangle,$
(A.7)

$Ell_{2}(M)=\left\langle\widehat{A}(TM)\mathrm{ch}\left(\Theta\left(T_{\mathbb{C}}M\right)\otimes\Theta_{2}\left(T_{\mathbb{C}}M\right)\right),[M]\right\rangle,$
(A.8)

$W(M)=\left\langle\widehat{A}(TM)\mathrm{ch}\left(\Theta\left(T_{\mathbb{C}}M\right)\right),[M]\right\rangle,$
(A.9)

where

$\displaystyle\begin{split}\Theta(T_{\mathbb{C}}M)&=\overset{\infty}{\underset{m=1}{\otimes}}S_{q^{m}}(\widetilde{T_{\mathbb{C}}M}),\\\
\Theta_{1}(T_{\mathbb{C}}M)&=\bigotimes_{m=1}^{\infty}\Lambda_{q^{m}}(\widetilde{T_{\mathbb{C}}M}),\\\
\Theta_{2}(T_{\mathbb{C}}M)&=\bigotimes_{m=1}^{\infty}\Lambda_{-q^{m-{1\over
2}}}(\widetilde{T_{\mathbb{C}}M})\end{split}$ (A.10)

are the Witten bundle introduced in [41].

By the Atiyah-Singer index theorem, when the manifold $M$ is spin, the
elliptic genera can be expressed analytically as index of the twisted Dirac
operators

$Ell_{1}(M)=\operatorname{ind}(d_{s}\otimes(\Theta\left(T_{\mathbb{C}}M\right)\otimes\Theta_{1}\left(T_{\mathbb{C}}M\right))\in\mathbb{Z}[[q]],$
$Ell_{2}(M)=\operatorname{ind}(D\otimes(\Theta\left(T_{\mathbb{C}}M\right)\otimes\Theta_{2}\left(T_{\mathbb{C}}M\right))\in\mathbb{Z}[[q^{1/2}]],$

and further when $M$ is spin, Witten genus can be expressed analytically as
index of the twisted Dirac operators,

$W(M)=\operatorname{ind}(D\otimes\Theta\left(T_{\mathbb{C}}M\right))\in\mathbb{Z}[[q]],$

where $d_{s}$ is the signature operator and $D$ is the Atiyah-Singer spin
Dirac operator on $M$ (cf. [19]).

Physically, they are partition functions of two-dimensional quantum field
theories [41] and formally the indices of (twisted) Dirac operator in free
loop spaces [40].

The purely topological definition of elliptic genera were given by Ochanine
[36] and Landweber-Stong [31] The theory of elliptic genera and Witten genus
lead to the profound theory of elliptic cohomology [22] in algebraic topology.

Elliptic genera and Witten genus have the following modularity properties [19,
28, 43].

######  Lemma .

(i) $Ell_{1}(M)$ and $Ell_{2}(M)$ are level 2 modular forms of weight $2m$ and
they are modularly related as

$Ell_{1}(M,-1/\tau)={(2\tau)}^{2m}Ell_{2}(M,\tau).$ (A.11)

(ii) $W(M)$ is a modular form of weight $2m$, when $M$ is string (i.e. spin
and the spin class $\frac{p_{1}(M)}{2}=0$).

Let

$\Theta_{2}\left(T_{\mathbb{C}}M\right)=B_{0}(T_{\mathbb{C}}M)+B_{1}(T_{\mathbb{C}}M)q^{1\over
2}+\cdots,$ (A.12)

where the $B_{i}$’s, are elements in the semi-group formally generated by
Hermitian vector bundles over $M$. Then

$Ell_{2}(M)=\operatorname{ind}(D\otimes
B_{0}(T_{\mathbb{C}}M))+\operatorname{ind}(D\otimes
B_{1}(T_{\mathbb{C}}M))q^{1/2}+\cdots.$ (A.13)

######  Lemma .

$Ell_{2}(M)$ is determined by $\operatorname{ind}(D\otimes
B_{k}(T_{\mathbb{C}}M)),0\leq k\leq\frac{m}{2}$, where
$B_{k}(T_{\mathbb{C}}M)$ involves linear combinations of tensor product of
$TM$ at most to power ${[\frac{m}{2}]}$.

###### Proof.

See [28]. ∎

Let

$\Theta\left(T_{\mathbb{C}}M\right)=W_{0}(T_{\mathbb{C}}M)+W_{1}(T_{\mathbb{C}}M)q+\cdots,$
(A.14)

where the $W_{i}$’s, are elements in the semi-group formally generated by
Hermitian vector bundles over $M$. Then

$W(M)=\operatorname{ind}(D\otimes
W_{0}(T_{\mathbb{C}}M))+\operatorname{ind}(D\otimes
W_{1}(T_{\mathbb{C}}M))q+\cdots.$ (A.15)

######  Lemma .

Let $(M,g)$ be string $4m$ dimensional Riemannian manifold, then $W(M)$ is
determined by finite many (depending only on $m$) $\operatorname{ind}(D\otimes
W_{j}(T_{\mathbb{C}}M))$.

###### Proof.

See [19]. Actually let

$\begin{split}&E_{4}(\tau)=1+240(q+9q^{2}+28q^{3}+\cdots),\\\
&E_{6}(\tau)=1-504(q+33q^{2}+244q^{3}+\cdots)\\\ \end{split}$ (A.16)

be the Eisenstein series. Their weights are $4,6$ respectively, algebraically
independent and freely generate $\mathcal{M}_{\mathbb{R}}(SL(2,\mathbb{Z})).$

Let

$\Theta\left(T_{\mathbb{C}}M\right)=W_{0}(T_{\mathbb{C}}M)+W_{1}(T_{\mathbb{C}}M)q+\cdots,$
(A.17)

where the $W_{i}$’s, are elements in the semi-group formally generated by
Hermitian vector bundles over $M$. Then

$W(M)=\operatorname{ind}(D\otimes
W_{0}(T_{\mathbb{C}}M))+\operatorname{ind}(D\otimes
W_{1}(T_{\mathbb{C}}M))q+\cdots.$ (A.18)

Comparing $q$-coefficients, we see

$W(M)=\sum_{4i+6j=2m,\ i,j\geq 0}a_{ij}E_{4}(\tau)^{i}E_{6}(\tau)^{j},$

where $a_{ij}$ is determined by finite many (depending only on $m$)
$\operatorname{ind}(D\otimes W_{j}(T_{\mathbb{C}}M))$. ∎

One of most striking properties about elliptic genus is rigidity. Let $M$ be a
closed smooth manifold and $P$ be a Fredholm operator on $M$. We assume that a
compact connected Lie group $G$ acts on $M$ nontrivially and that $P$ is
$G$-equivariant, by which we mean it commutes with the $G$ action. Then the
kernel and cokernel of $P$ are finite dimensional representations of $G$. The
equivariant index of $P$ is the virtual character of $G$ defined by

$\operatorname{ind}(h,P)=\operatorname{Tr}\big{[}h\big{|}_{\ker
P}\big{]}-\operatorname{Tr}\big{[}h\big{|}_{\operatorname{coker}P}\big{]},\quad
h\in G.$ (A.19)

$P$ is said to be _rigid_ for this $G$ action if $\operatorname{ind}(h,P)$
does not depend on $h\in G$. Motivated by physics, Witten conjectured that the
operators
$d_{s}\otimes(\Theta\left(T_{\mathbb{C}}M\right)\otimes\Theta_{1}\left(T_{\mathbb{C}}M\right)$
and
$D\otimes(\Theta\left(T_{\mathbb{C}}M\right)\otimes\Theta_{2}\left(T_{\mathbb{C}}M\right)$
should be rigid. The Witten conjecture was first proved by Taubes [39] and
Bott-Taubes [7]. In [28, 29], using the modular invariance property, Liu
presented a simple and unified proof as well as various generalizations of the
Witten conjecture.

A celebrated result asserts that a string manifold with a nontrivial
$S^{3}$-action has vanishing Witten genus, see [28] and [13]. On the other
hand, concerning the application to positive Ricci curvature, Stolz
conjectures that if $M$ is a smooth closed string manifold of dimension $4m$
admitting a Riemannian metric with positive Ricci curvature, then the Witten
genus $W(M)$ vanishes [38]. So far the conjecture is still open.

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# Kinematics and Dynamics Modeling of 7 Degrees of Freedom Human Lower Limb
Using Dual Quaternions Algebra

Zineb BENHMIDOUCH, Saad MOUFID, Aissam AIT OMAR
<EMAIL_ADDRESS>

###### Abstract

Denavit and Hartenberg based methods as Cardan, Fick and Euler angles describe
the position and orientation of an end-effector in Three Dimensional (3D)
space. However, the generation of unrealistic human posture in joint space
constitutes the weak point to these methods because they impose a well-defined
rotations order. A method to handle the transformation homogeneous performance
uses the dual quaternions. Quaternions have proven themselves in many fields
as providing a computational efficient method to represent a rotation, and
yet, they can not deal with the translations in 3D-space. The dual numbers can
extend quaternions to dual quaternions. This paper exploits dual quaternions
theory to provide a fast and accurate solution to the forward, inverse
kinematics and recursive Newton-Euler dynamics algorithm for 7 Degree of
Freedom (DOF) human lower limb in 3D-space.

_K_ eywords Dual numbers $\cdot$ Quaternions $\cdot$ Dual quaternions algebra
$\cdot$ Artificial neural networks

##  1 Introduction

Compared to classical methods as Cardan , Fick and Euler angles which are
based on homogeneous transformation, dual quaternions [1] offer an
advantageous representation of a rigid transformation in 3D-space in many
aspects. The dual quaternions have less computer memory cost, computer memory
locations, since it needs 8 elements while classical methods require 12
elements, to describe a rotation composed with translation of a rigid body in
3D-space. Thus, the homogeneous transformation methods cost more storage and
require a high computational time due to the nonlinearity of the end-effector
coordinates. Moreover, these methods impose a well-defined rotation order.
While, the dual quaternions method enables the reduction of the number of
mathematical operations which leads automatically to minimize the
computational cost [2]. Moreover, dual quaternions method permits to avoid
discontinuities and singularities that arise from the Euler angle
representation based on cylindrical polar coordinates to represent the motion
of a rigid body in 3D-space [3].
Further, the dual quaternions method allows to avoid the gimbal lock
phenomenon. A gimbal lock problem is a loss of 1-DOF in 3D-space that occurs
when using the Euler angles [4]. As an example, suppose a rigid body rotates
in 3D-space in the following order Z, Y and X and the angle of rotation about
the axis Y equal to 90°. Thus, the first and second rotations are done
correctly, while the rotation around the axis X is impossible because the
three rotations axis become coplanar.
Dual quaternions algebra benefits have been argued in many works comprising
dynamics modeling of a mobile manipulator [5], stabilization of a rigid body
motion, multiple body interaction [6], inverse kinematic study of 6-DOF of
robot arm and tracking control [7, 8], etc. Valverde et al. [9] presented a
serial manipulator dynamics using the recursive Newton-Euler based on dual
quaternions. While, Silva et al. [10] used recursive Newton-Euler and Gauss’s
Principle of Least Constraint based on dual quaternions to describe the
relations between the joint velocities, forces and torques variables of a
mobile manipulators.
In this paper, the dual quaternions based theory was applied for the
kinematics and dynamics study of the 7-DOF human lower limbs in 3D-space.
Thereafter, the artificial neural networks method was used to solve the
inverse kinematics problem. The efficiency of the artificial neural networks
method were verified by using the jerk energy criteria. The rest of this paper
is organized as follows. Section 2 provides a brief mathematical theory
background on dual quaternions algebra, Section 3 elaborates on the forward
kinematics of the human lower limb in 3D-space by using the dual quaternions.
While Section 4 focuses on the application of the artificial neural network
method to solve the inverse kinematics of the lower limb. In section 5, the
dynamical model of lower limb using dual quaternion based on a recursive
Newton-Euler is developed. Finally, in sections 6 the simulation results are
discussed.

##  2 PRELIMINARY THEORY

This section provides a theory background on dual quaternions required to
describe a rigid body motion in 3D-space. The rotation matrix
$\mathbb{R}^{3\times 3}$ is an orthogonal matrix defined by the Lie group
$SO(3)\subset O(3)$, as follows:

$SO(3)=\left\\{R\in\mathbb{R}^{3\times 3},R^{T}R=I_{3},det(R)=\pm 1\right\\}$
(1)

Where the orthogonal groups $O(3)$ and $SO(3)$ consist of the endomorphisms
that preserves the euclidean norm and the normal direct isometries of $O(3)$,
respectively. Thus, $SO(3)$ group preserves the orientation of the space [11].
The Euler angles convention is a common method to represent the special
orthogonal group $SO(3)$. However, it shows a singularity at
$\pm(n+1)\frac{\pi}{2}$, for all $n\in\mathbb{N}$. Further, at certain
configurations there are infinite possible solutions, sequence of rotations
about the three axes, for the mapping from the special orthogonal group
$SO(3)$ to Euler angles $\xi$ [12].
Euler’s Theorem on the axis of a 3D-space Rotation states if $R\in SO(3)$,
then there is a non-zero vector $u$ that satisfy $Ru=u$. That means, the
relative rotation of any two reference frames is equivalent to a single
rotation about a given vector $u$ with a given rotation angle. Therefore, the
orientation of a rigid body in 3D-space can be described by a quaternion [13]:

$q=q_{0}+q_{1}i+q_{2}j+q_{3}k=\cos(\frac{\alpha}{2})+\sin(\frac{\alpha}{2})v$
(2)

where $v$ and $\alpha$ are the rotation unit vector and angle, respectively.
Multiplication of the quaternion components is done by using the following
rules:

$\begin{array}[]{c}i^{2}=j^{2}=k^{2}=ijk=-1\\\ ij=-ji=k\\\ jk=-kj=i\\\
ki=-ik=j\end{array}$ (3)

Let a quaternion $q$ represent the rotation of the vector
$\vec{p}\in\mathbb{R}^{3}$ in 3D-space, thus the adjoint transformation is
given by:

$\vec{p^{\prime}}=Ad(q)\vec{p}=q\vec{p}q^{*}$ (4)

where $\vec{p^{\prime}}\in\mathbb{R}^{3}$ is the resulting vector.
Quaternions can only deal with rotations of a rigid body around a fixed point.
In order to also consider the translations, the dual numbers notion will be
used. The concept of dual numbers, also called duplex, were first introduced
by Clifford in 1873 [14], is a powerful algebraic structure for the spatial
mechanisms kinematic and dynamic analysis. Dual numbers are of the form
$a+\epsilon b\in\mathbb{D}$, where $\epsilon$ is a new element called
nilpotent number such that $\epsilon^{2}=\epsilon^{3}=...=\epsilon^{n}=0$,
where $n\in\mathbb{N}$. Dual quaternions are an extension of quaternions using
dual numbers concept expressed as follows:

$D=q_{0}+\epsilon q_{1}$

Where $q_{0}$ and $q_{1}$ are quaternions denote the primary and dual parts of
$D$, and $\epsilon$ represents the dual unit. The conjugate of $D$ is defined
as:

$D^{*}=q_{0}^{*}-\epsilon q_{1}^{*}$ (5)

Therefore, the dual quaternions are used to describe a pure rotation of a
rigid body in space as follows:

$D_{R}=c(\frac{\alpha}{2})+s(\frac{\alpha}{2})v$ (6)

where $\alpha$ and $v$ represent the rotation angle and unit vector,
respectively. Hestenes [15], shows that the derivative of the rotation dual
quaternion is given by the following equation:

$\dot{D}_{R}=\frac{1}{2}D_{R}\vec{\Omega}_{R}$ (7)

Where $\vec{\Omega}_{R}$ denotes the dual quaternion of the rotation velocity.
Further, the derivative of the conjugate of the rotation dual quaternion is
given by:

$\dot{D}_{R}^{*}=-\frac{1}{2}\vec{\Omega}_{R}D_{R}^{*}$ (8)

Furthermore, a pure translation is described by the following form:

$D_{T}=1+\epsilon\frac{d}{2}t$ (9)

Where $t$ and $d$ represent the unitary axis coordinates and the distance of
the translation, respectively. Also, the translation conjugate is equal to
itself. Whence, the velocity of the translation is given by:

$\dot{D}_{T}=\epsilon\frac{\dot{d}}{2}t+\epsilon\frac{d}{2}\dot{t}$ (10)

The dual quaternion of the position and the velocity of a point $P$ are as
follows:

$\begin{array}[]{c}P=1+\epsilon p\\\ \dot{P}=\epsilon\dot{p}\end{array}$ (11)

where $p$ is the coordinates of $P$ in 3D-space.

##  3 Forward kinematics for lower limbs

In this section, the Forward Kinematics (FK) of the lower limbs, depicted in
Figure 1, using the dual quaternions is established. FK refers to compute the
positions and orientations of the end-effector in task space from the axis and
the angles of the joint rotations. However, the lower limb is decomposed into
four segments, the pelvis, the thigh, the leg and the foot, linked together by
three joint groups, the hip which rotates about three perpendicular axes, the
knee which moves only about z-axis and the ankle permits movement in three
planes. Thus, DOF of the lower limbs is equal to 7 [16]. Therefore, the end-
effector position relative to the reference frame $\mathscr{R}_{3}$, is
denoted by $P_{E/3}$, can be expressed as:

$P_{E/3}=Ad(r_{3})Ad(t_{23})Ad(r_{2})Ad(t_{12})Ad(r_{1})Ad(t_{01})P_{E/0}=r_{3}t_{23}r_{2}t_{12}r_{1}t_{01}P_{E/0}t_{01}r_{1}^{*}t_{12}r_{2}^{*}t_{23}r_{3}^{*}$
(12)

Where the numbers $1,~{}2$ and $3$ stand for hip, knee and ankle,
respectively.
$P_{E/3}=1+\epsilon\frac{1}{2}\left(\begin{array}[]{ccc}L_{3}&0&0\end{array}\right)^{T}$
and $P_{E/0}$ are the end-effector position relative to reference frame
$\mathscr{R}_{3}$ and $\mathscr{R}_{0}$, respectively.

$\begin{array}[]{c}r_{1}=c(\frac{\theta_{1}}{2})+n_{1}s(\frac{\theta_{1}}{2})\\\
r_{2}=c(\frac{\theta_{2}}{2})+n_{2}s(\frac{\theta_{2}}{2})\\\
r_{3}=c(\frac{\theta_{3}}{2}+\frac{\pi}{2})+n_{3}s(\frac{\theta_{3}}{2}+\frac{\pi}{2})\end{array}$
(13)

Equations (13) represent the thigh rotation around the pelvis, the leg
rotation around the thigh, the foot rotation around the leg, respectively.
where $n_{1}$, $n_{2}$ and $n_{3}$ are the rotation axis given by:

$\begin{array}[]{c}n_{1}=\left(\begin{array}[]{ccc}n_{1x}&n_{1y}&n_{1z}\end{array}\right)^{T}\\\
n_{2}=\left(\begin{array}[]{ccc}0&0&1\end{array}\right)^{T}\\\
n_{3}=\left(\begin{array}[]{ccc}n_{3x}&n_{3y}&n_{3z}\end{array}\right)^{T}\end{array}$
(14)

The translation dual quaternions representing the offset between the pelvis
center and the hip, the offset between the hip and the knee and the offset
between the knee and the ankle, respectively, are given by:

$\begin{array}[]{c}t_{01}=1+\epsilon\frac{1}{2}\left(\begin{array}[]{ccc}0&0&L_{0}\end{array}\right)^{T}\\\
t_{12}=1+\epsilon\frac{1}{2}\left(\begin{array}[]{ccc}L_{1}&0&0\end{array}\right)^{T}\\\
t_{23}=1+\epsilon\frac{1}{2}\left(\begin{array}[]{ccc}L_{2}&0&0\end{array}\right)^{T}\end{array}$
(15)

Therefore, the forward kinematics can be obtained using the following
equation:

$P_{E/0}=t_{01}r_{1}^{*}t_{12}r_{2}^{*}t_{23}r_{3}^{*}P_{E/3}r_{3}t_{23}r_{2}t_{12}r_{1}t_{01}$
(16)

From Equation (12):

$P_{E/3}=r_{3}P_{E/2^{\prime}}r_{3}^{*}$ (17)

Where $P_{E/2^{\prime}}$ is the coordinates of the end-effector relative to
reference frame $\mathscr{R}_{2}^{\prime}$. Thereafter, the end-effector
velocity relative to reference frame $\mathscr{R}_{3}$ is given by the
following expression:

$\dot{P}_{E/3}=\dot{R}_{3}P_{E/2^{\prime}}R_{3}^{*}+R_{3}\dot{P}_{E/2^{\prime}}R_{3}^{*}+R_{3}P_{E/2^{\prime}}\dot{R}_{3}^{*}$
(18)

Using Equations (8) and (9), $\dot{P}_{E/3}$ is given as follows:

$\dot{P}_{E/3}=R_{3}\left(\frac{1}{2}\left(\Omega_{3}P_{E/2^{\prime}}-P_{E/2^{\prime}}\Omega_{3}\right)+\dot{P}_{E/2^{\prime}}\right)R_{3}^{*}$
(19)

Where $\Omega_{3}$ is the rotation vector velocity of the foot around the leg.
Hence:

$\dot{P}_{E/2^{\prime}}=\dot{P}_{E/2}$ (20)

In the same way, the end-effector velocities relative to reference frame
$\mathscr{R}_{2}$ and $\mathscr{R}_{1}$ are given as follows:

$\dot{P}_{E/2}=R_{2}\left(\frac{1}{2}\left(\Omega_{2}P_{E/1^{\prime}}-P_{E/1^{\prime}}\Omega_{2}\right)+\dot{P}_{E/1^{\prime}}\right)R_{2}^{*}$
(21)
$\dot{P}_{E/1}=R_{1}\left(\frac{1}{2}\left(\Omega_{1}P_{E/0^{\prime}}-P_{E/0^{\prime}}\Omega_{1}\right)+\dot{P}_{E/0}\right)R_{1}^{*}$
(22)

Where $\Omega_{2}$ and $\Omega_{1}$ are the rotation vector velocities of the
leg around the thigh and the thigh around the pelvis, respectively. Therefore,
the final expression of the end-effector velocity relative to reference frame
$\mathscr{R}_{0}$ is given as follows:

$\dot{P}_{E/0}=\frac{1}{2}\left(P_{E/0}\Omega_{1}-\Omega_{1}P_{E/0}\right)+\frac{1}{2}R_{1}^{*}\left(P_{E/1^{\prime}}\Omega_{2}-\Omega_{2}P_{E/1^{\prime}}\right)R_{1}+\frac{1}{2}R_{1}^{*}R_{2}^{*}\left(P_{E/2^{\prime}}\Omega_{3}-\Omega_{3}P_{E/2^{\prime}}\right)R_{2}R_{1}$
(23)

Figure 1: Physiological lower limbs diagram

##  4 Inverse kinematics for lower limbs

The Inverse Kinematics (IK) problem provides each joint angles from the known
end-effector position and velocity in order to achieve an efficient control of
the lower limb motion. However, the lower limb has too many DOF compared to
its dimension which leads to the redundancy problem. Therefore, the IK problem
of the lower limb is complicated since it has an infinite solutions in
3D-space. However, in the following a neural networks method will be proposed
to handle the redundancy problem and yield one optimal solution.
Thereafter, the neural networks is trained by exploiting the data delivered by
the Equations (16) and (23) to learn the angular and rotation axis joint
values of the configuration space. In other words, this method allows to solve
this nonlinear IK by identifying which joint configuration
$\theta_{1},\theta_{2}$ and $\theta_{3}$ and rotation axis
$n_{1}=(n_{1x},n_{1y},n_{1z})^{T}$ and $n_{3}=(n_{3x},n_{3y},n_{3z})^{T}$
correspond to the predefined end-effector position $(x_{d},y_{d},z_{d})^{T}$
and velocity $(\dot{x}_{d},\dot{y}_{d},\dot{z}_{d})^{T}$.
In order to create an efficient database, the workspace constraints must be
respected. So, the rotation axis constraints are given as follows:

$\bigg{\\{}\begin{array}[]{c}n_{3x}^{2}+n_{3y}^{2}+n_{3z}^{2}=1\\\
n_{1x}^{2}+n_{1y}^{2}+n_{1z}^{2}=1\end{array}$ (24)

Moreover, the hip and knee joints have the mechanical characteristic of
spherical joint, move in three planes, while the knee joint form a pivot
joint. However, these movements are limited by the upper and lower values as
shown in Table 1. So, the following inequalities allows to generate a set of
angular joint values while respecting the hip, knee and ankle range of motion:

$\begin{array}[]{c}-30^{\circ}\leqslant
arccos\left(\frac{P_{C/0}-P_{B/0}}{|P_{C/0}-P_{B/0}|}\cdot\vec{x_{0}}\right)\leqslant
120^{\circ}\\\ -20^{\circ}\leqslant
arccos\left(\frac{P_{C/0}-P_{B/0}}{|P_{C/0}-P_{B/0}|}\cdot\vec{z_{0}}\right)\leqslant
45^{\circ}\\\ -50^{\circ}\leqslant
arccos\left(\frac{P_{C/0}-P_{B/0}}{|P_{C/0}-P_{B/0}|}\cdot\vec{y_{0}}\right)\leqslant
40^{\circ}\\\ -150^{\circ}\leqslant\theta_{2}\leqslant 0^{\circ}\\\
-40^{\circ}\leqslant
arccos\left(\frac{P_{E/0}-P_{D/0}}{|P_{E/0}-P_{D/0}|}\cdot\vec{x_{0}}\right)\leqslant
20^{\circ}\\\ -35^{\circ}\leqslant
arccos\left(\frac{P_{E/0}-P_{D/0}}{|P_{E/0}-P_{D/0}|}\cdot\vec{z_{0}}\right)\leqslant
30^{\circ}\\\ -35^{\circ}\leqslant
arccos\left(\frac{P_{E/0}-P_{D/0}}{|P_{E/0}-P_{D/0}|}\cdot\vec{y_{0}}\right)\leqslant
20^{\circ}\\\ \end{array}$ (25)

Table 1: Range of motion of lower limb Joint | Movements | Range of motion
---|---|---
Hip | Flexion/Extension | From $-30^{\circ}$ to $120^{\circ}$
| Adduction/Abduction | From $-20^{\circ}$ to $45^{\circ}$
| Medial/Lateral rotation | From $-50^{\circ}$ to $40^{\circ}$
Knee | Flexion/Extension | From $-150^{\circ}$ to $0^{\circ}$
Ankle | Plantarflexion/Dorsiflexion | From $-40^{\circ}$ to $20^{\circ}$
| Pronation/External rotation | From $-35^{\circ}$ to $30^{\circ}$
| Inversion/Eversion | From $-35^{\circ}$ to $20^{\circ}$

##  5 Dynamics modeling of lower limb

This section focuses on the dynamical description of the lower limb shown in
Figure 2 using Dual quaternion based recursive Newton-Euler method. That
means, given the corresponding lower limb configuration, namely, positions,
velocities and accelerations, the link’s center of mass velocities and
accelerations, formed the twist, are calculated based on Newton Euler
propagation law. Thereafter, the wrenches which are a representation of forces
and moments acting on each link in 3D-space, are deduced starting from the
wrenches applied to the end effector.

Figure 2: Lower limb with 7-DOF

###  5.1  Forward Recursion

The forward recursion aims at obtaining the twist, that is, the angular and
linear velocities of each link’s center of mass, and its first derivative. Let
$\xi_{0,c_{0}}^{c_{0}}$ the twist of the center of mass $C_{0}$ with respect
to frame $\mathscr{R}_{0}$, expressed in frame $\mathscr{R}_{c_{0}}$, and it
is given by the following dual quaternion:

$\xi_{0,c_{0}}^{c_{0}}=\omega_{0,c_{0}}^{c_{0}}+\epsilon
v_{0,c_{0}}^{c_{0}}=0$ (26)

where $\omega_{0,c_{0}}^{c_{0}}$ and $v_{0,c_{0}}^{c_{0}}$ are, respectively,
the angular and linear velocities. In general, the twist
$\xi_{0,c_{i}}^{c_{i}},~{}~{}i\in[1,3]$ expressed in reference frame
$\mathscr{R}_{c_{i}}$ that yields the motion of $\mathscr{R}_{c_{i}}$ with
respect to frame $\mathscr{R}_{0}$, is given by:

$\xi_{0,c_{i}}^{c_{i}}=\xi_{0,i}^{c_{i}}+\xi_{i,c_{i}}^{c_{i}}$ (27)

where, $\xi_{0,i}^{c_{i}}$ and $\xi_{i,c_{i}}^{c_{i}}$ are, respectively, the
twist related to the motion of $i^{th}$ joint and $i^{th}$ link’s center of
mass. The twist of the $i^{th}$ link’s center of mass with respect to the
reference frame $\mathscr{R}_{0}$ depends on the twist provided by by its
joint and on the twist of the $(i-1)^{th}$ relative to $i^{th}$ joint because
they are physically attached. Therefore, using Equation (4), the equation 27
can be expressed as follows:

$\xi_{0,c_{i}}^{c_{i}}=\xi_{0,i}^{c_{i}}+\xi_{i,c_{i}}^{c_{i}}=Ad(x_{c_{i-1}}^{c_{i}})(\xi_{0,c_{i-1}}^{c_{i-1}}+\xi_{c_{i-1},i}^{c_{i-1}})+Ad(x_{i}^{c_{i}})\xi_{i,c_{i}}^{i}$
(28)

where $x_{c_{i-1}}^{c_{i}}=r_{c_{i-1}}^{c_{i}}+\frac{1}{2}\epsilon
r_{c_{i-1}}^{c_{i}}p_{c_{i-1}}^{c_{i}}$ and
$x_{i}^{c_{i}}=r_{i}^{c_{i}}+\frac{1}{2}\epsilon r_{i}^{c_{i}}p_{i}^{c_{i}}$
are units dual quaternions where $r$ and $p$ denote rotation and position in
3D-space, respectively.As well as, $\xi_{c_{i-1},i}^{c_{i-1}}=0$ because
$x_{c_{i-1},i}^{c_{i-1}}$ is constant. Thus,

$\xi_{0,c_{i}}^{c_{i}}=Ad(x_{c_{i-1}}^{c_{i}})\xi_{0,c_{i-1}}^{c_{i-1}}+Ad(x_{i}^{c_{i}})\xi_{i,c_{i}}^{i}$
(29)

where,

$\xi_{i,c_{i}}^{i}=\omega_{i,c_{i}}^{i}+\frac{1}{2}\epsilon(\dot{P}_{i,c_{i}}^{i}+\omega_{i,c_{i}}^{i}\times
P_{i,c_{i}}^{i})$ (30)

is the twist of reference frame $\mathscr{R}_{c_{i}}$ with respect to
reference frame $\mathscr{R}_{i}$, expressed in frame $\mathscr{R}_{i}$.
where,

$P_{i,c_{i}}^{i}\times\omega_{i,c_{i}}^{i}=\frac{P_{i,c_{i}}^{i}\omega_{i,c_{i}}^{i}-\omega_{i,c_{i}}^{i}P_{i,c_{i}}^{i}}{2}$
(31)

Therefore, the time derivative of Equation (29) is given as follows:

$\begin{split}\dot{\xi}_{0,c_{i}}^{c_{i}}=Ad(x_{c_{i}-1}^{c_{i}})\dot{\xi}_{0,c_{i}-1}^{c_{i}-1}+Ad(x_{i}^{c_{i}})\dot{\xi}_{i,c_{i}}^{i}+\xi_{c_{i},c_{i}-1}^{c_{i}}\times(Ad(x_{c_{i}-1}^{c_{i}})\xi_{0,c_{i}-1}^{c_{i}-1})\end{split}$
(32)

###  5.2 Backward Recursion

Backward recursion consists in computing the required joints torques to obtain
the predefined motion from the dual quaternion wrenches acting at the end
effector and the twists and their first time derivative of each link’s center
of the mass.
The wrench at the foot’s center of mass is given by:

$\zeta_{0,c_{3}}^{c_{3}}=f_{0,c_{3}}^{c_{3}}+m_{3}g^{c_{3}}+\epsilon\tau_{0,c_{3}}^{c_{3}}$
(33)

where $g^{c_{3}}$ is the force of gravity and $f_{0,c_{3}}^{c_{3}}$ is the
force applied at the foot’s center of mass. Moreover, $\tau_{0,c_{3}}^{c_{3}}$
is the torque about the the foot’s center of mass results of its angular
momentum variation, given by:

$\tau_{0,c_{3}}^{c_{3}}=\chi_{3}(I_{3}^{c_{3}})\mathcal{P}(\dot{\xi}_{0,c_{3}}^{c_{3}})+\mathcal{P}(\xi_{0,c_{3}}^{c_{3}})\times\chi_{3}(I_{3}^{c_{3}})\mathcal{P}(\xi_{0,c_{3}}^{c_{3}})$
(34)

where $\mathcal{P}$ return the primary component,
$I_{3}^{c_{3}}=(i_{x},i_{y},i_{z})\in\mathbb{H}^{3}$ is the quaternionic
inertia tensor and $\chi_{3}$ is the operator defined as

$\chi_{3}(I)\mathcal{P}=<i_{x},\mathcal{P}>i+<i_{y},\mathcal{P}>j+<i_{z},\mathcal{P}>k$
(35)

where $<.,.>:\mathbb{H}\rightarrow\mathbb{R},~{}~{}<a,b>=-\frac{ab+ba}{2}$ is
the inner product. Using the adjoint transformation, the wrench
$\zeta_{0,3}^{c_{3}}$ is given by:

$\zeta_{0,3}^{c_{3}}=Ad(x_{c_{3}}^{3})\zeta_{0,c_{3}}^{c_{3}}$ (36)

Thereafter, The wrench $\zeta_{0,i}^{c_{i}},~{}~{}i\in[1,2]$ of $i^{th}$ joint
is given by:

$\zeta_{0,i}^{c_{i}}=Ad(x_{c_{i}}^{i})\zeta_{0,c_{i}}^{c_{i}}+Ad(x_{i+1}^{i})\zeta_{0,i+1}^{c_{i+1}}$
(37)

where
$\zeta_{0,c_{i}}^{c_{i}}=f_{0,c_{i}}^{c_{i}}+m_{i}g^{c_{i}}+\epsilon\tau_{0,c_{i}}^{c_{i}}$
is the wrench at the $i^{th}$ link’s center of mass.
Extraction of $\tau_{c_{i}}^{c_{i}}$ components of the wrench
$\zeta_{0,c_{i}}^{c_{i}}$ yield the dynamic equation of the lower limb that
can be written in the following form:

$\tau=M(q)\ddot{q}+V(q,\dot{q})+G(q)$ (38)

where $M(q)$ is the mass matrix of the lower limb, $V(q,\dot{q})$ is the
centrifugal and Coriolis vector, and $G(q)$ is the gravity force vector.

Figure 3: Trajectory planning for the lower limb in 3D-space Figure 4: ANN
algorithm

(a) Rotation vectors coordinates

(b) Articulation angle values

Figure 5: Simulation results Figure 6: Position Error  Figure 7: Desired and
calculated trajectory in 3D-space Figure 8: Energy of the motion

##  6 Simulation results

First of all, the set of waypoints to generate a trajectory in task space is
supposed available from the task planner. Therefore, the trajectory is planned
by using the minimum jerk criterion to assume the continuity of the
acceleration of the joints that can leads to reduce vibrations and to avoid
resonances frequencies. For this simulation example, the position, velocity,
acceleration values at the beginning and at the end of motion are shown in
Table 2. Therefore, the trajectory planning simulation result is shown in
Figure 3.

Table 2: Initial and final position, velocity, acceleration of a motion values | Initial value | Final value
---|---|---
Position ($m$) | $(0.8,-0.06,0.1)$ | $(0.7,0.48,0.08)$
Speed ($m/s$) | 0 | 0
Acceleration ($m/s^{2}$) | 0 | 0

Thereafter, the IK of the lower limb is obtained by using a multi-layer
perceptron trained with Levenberg-Marquardt back propagation algorithm by
using $400000$ samples collected, where its structure is given by Figure 4. It
has a two-layer feed-forward network, namely, a hidden layer with 20
interconnected sigmoid normal neurons and an output layer with 9 linear output
neurons. The simulation results are depicted in Figure 5 where the
computational time is equal to $0.009s$. While, Figure 6 depict the position
error along x axis, y-axis and z-axis, respectively. Where the root mean
square error along x axis, y-axis and z-axis are $0.0741$, $0.0970$ and
$0.0776$, respectively. The position error between desired and calculated
trajectory shown in Figure 7 is almost negligible. This confirms that this
method performs well from the point of view of accuracy. Moreover, human
should not produce a high energy during the motion in order to overcome the
fatigue. While, the energy constraint of the angular joints can be considered
as the third derivative due to the fact that the energy is related to the
minimum jerk criterion. Therefore, from Figure 8, the energy sum of
$\theta_{1}$, $\theta_{2}$ and $\theta_{3}$ are less than $0.4~{}J$.

##  7 Conclusion

The main focus of this paper was to use the dual quaternions algebra to
describe the kinematics, that is, the position and orientation, and dynamics
modeling of an anthropomorphic leg in 3D-space in order to avoid the high
costs of the homogeneous transformation methods. Therefore, ANN were used to
solve the IK problem while respecting the range of motion constraints and the
minimum energy criterion was used to prove the realistic human posture. While,
Newton-Euler recursive method based on dual quaternions was opted for the
dynamics modeling in order to avoid the geometrical analyses complexity.

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